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<span class="kw">set </span><font color="Maroon" title="c30">a</font> =  <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c31">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <br/> for <font color="Olive" title="b3">k0</font>, <font color="Olive" title="b4">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Olive" title="b3">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c31">s</font>,<font color="Olive" title="b3">k0</font> &amp; <font color="Olive" title="b2">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c31">s</font>,<font color="Olive" title="b3">k0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b1">f</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Olive" title="b4">k</font> holds <br/>( <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b2">g</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Olive" title="b4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> ) &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b6">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">j</font> &amp; <font color="Olive" title="b6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">j</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c32">f</font>, <font color="Maroon" title="c33">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k0</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Olive" title="b1">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c32">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c31">s</font>,<font color="Olive" title="b1">k0</font> &amp; <font color="Maroon" title="c33">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c31">s</font>,<font color="Olive" title="b1">k0</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Maroon" title="c32">f</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Olive" title="b2">k</font> holds <br/>( <font color="Maroon" title="c32">f</font>,<font color="Maroon" title="c33">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c33">g</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> holds <br/><font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c34">m</font>, <font color="Maroon" title="c35">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c32">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c31">s</font>,<font color="Maroon" title="c34">m</font> &amp; <font color="Maroon" title="c33">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c31">s</font>,<font color="Maroon" title="c34">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> &amp; <font color="Maroon" title="c32">f</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> implies ( <font color="Maroon" title="c32">f</font>,<font color="Maroon" title="c33">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c33">g</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> holds <br/><font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:37"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
( <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c32">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c31">s</font>,<font color="Maroon" title="c34">m</font> or  not <font color="Maroon" title="c33">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c31">s</font>,<font color="Maroon" title="c34">m</font> or  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">g</font> or  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> or  not <font color="Maroon" title="c32">f</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> or ( <font color="Maroon" title="c32">f</font>,<font color="Maroon" title="c33">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c33">g</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> holds <br/><font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <br/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b2">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c34">m</font> &amp; <font color="Olive" title="b3">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c34">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">f2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> $1 &amp; <font color="Olive" title="b2">f1</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,$1 holds <br/>( <font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b3">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f1</font> holds <br/><font color="Olive" title="b2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> ) &amp; ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> ) ) );<br/>
<span class="kw">assume </span><a NAME="E2:37"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
( <font color="Maroon" title="c32">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c31">s</font>,<font color="Maroon" title="c34">m</font> &amp; <font color="Maroon" title="c33">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c31">s</font>,<font color="Maroon" title="c34">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">g</font> or  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> or  not <font color="Maroon" title="c32">f</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> or ( <font color="Maroon" title="c32">f</font>,<font color="Maroon" title="c33">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c33">g</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> holds <br/><font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) ) )</span><br/>

<div><span class="lab"><font color="Green" title="E27">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c36">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c36">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:37_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c36">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c37">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b1">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c37">t</font>,<font color="Maroon" title="c34">m</font> &amp; <font color="Olive" title="b2">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font>,<font color="Maroon" title="c34">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">f1</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>( <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Olive">b<sub>5</sub></font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive">b<sub>6</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>6</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>5</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>6</sub></font> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>5</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>7</sub></font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c38">f1</font>, <font color="Maroon" title="c39">f2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c38">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c37">t</font>,<font color="Maroon" title="c34">m</font> &amp; <font color="Maroon" title="c39">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font>,<font color="Maroon" title="c34">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">f2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c38">f1</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>5</sub></font> ) ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:37_1_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:37_1_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:37_1_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E5:37_1_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c38">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c37">t</font>,<font color="Maroon" title="c34">m</font>
 <span class="kw">and </span><br/><a NAME="E6:37_1_1"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c39">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font>,<font color="Maroon" title="c34">m</font>
 <span class="kw">and </span><br/><a NAME="E7:37_1_1"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">f2</font>
 <span class="kw">and </span><br/><a NAME="E8:37_1_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E9:37_1_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c38">f1</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>5</sub></font> ) ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c40">Bt</font> =  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font>;<br/><span class="kw">set </span><font color="Maroon" title="c41">x</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c42">y</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><a NAME="E10:37_1_1"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="scpisort.html#E23"><span class="lab"><font color="Green" title="E15">Lm7</font></span></a></span>;<br/><a NAME="E11:37_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a></span>;<br/><a NAME="E12:37_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="lab"><font color="Green" title="E40">A16</font></span>:  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> = 
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c34">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a></span>
<br/>.= 
 <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">k</font> <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="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></span>
;<br/><a NAME="E14:37_1_1"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
( <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="scpisort.html#E24"><span class="lab"><font color="Green" title="E16">Lm8</font></span></a></span>;<br/><a NAME="E15:37_1_1"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
( <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="scpisort.html#E24"><span class="lab"><font color="Green" title="E16">Lm8</font></span></a></span>;<br/><a NAME="E16:37_1_1"/><span class="lab"><font color="Green" title="E43">A19</font></span>: 
 <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></span>;<br/><a NAME="E17:37_1_1"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="scpisort.html#E24"><span class="lab"><font color="Green" title="E16">Lm8</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><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="E1:37_1_1_2"/>
( <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> or <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>5</sub></font> ) ) )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c43">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">i</font> &amp; <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">f2</font> implies <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:37_1_1_2_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">i</font> &amp; <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">f2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span></span><br/><span class="kw">hence </span><font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="scpisort.html#E33"><span class="lab"><font color="Green" title="E23">Lm15</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:37_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<font color="Maroon" title="c39">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span>,<font color="Maroon" title="c34">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>;<br/><a NAME="E4:37_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c43">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <span class="kw"> such that </span><br/><a NAME="E6:37_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c43">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font>
 <span class="kw">and </span><br/><a NAME="E7:37_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c43">h</font> holds <br/><font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scpisort.html#T2" target="_self" title="SCPISORT:th.2">Th2</a></span>;<br/><a NAME="E8:37_1_1_2_1"/>
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c36">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E9:37_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c43">h</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c36">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E51">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> implies <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1_2_1_2"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
( <font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 <span class="kw">and </span><br/><a NAME="E2:37_1_1_2_1_2"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font>
 <span class="kw">and </span><br/><a NAME="E3:37_1_1_2_1_2"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font></span><br/><a NAME="E4:37_1_1_2_1_2"/><span class="lab"><font color="Green" title="E55">A31</font></span>: 
( <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_2"><span class="lab"><font color="Green" title="E52">A28</font></span></a></span>;<br/><a NAME="E5:37_1_1_2_1_2"/>
<font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1_2_1_2"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E6:37_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A32</font></span>: 
<font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E2:37_1_1_2_1_2"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E4:37_1_1_2_1_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="scpisort.html#E24"><span class="lab"><font color="Green" title="E16">Lm8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E2:37_1_1_2_1_2"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c44">i</font>, <font color="Maroon" title="c45">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font> &amp; <font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> implies <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1_2_1_3"/><span class="lab"><font color="Green" title="E52">A33</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_1_2_1_3"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">and </span><br/><a NAME="E3:37_1_1_2_1_3"/><span class="lab"><font color="Green" title="E54">A35</font></span>: 
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font></span><br/><a NAME="E4:37_1_1_2_1_3"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1_3"><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="E5:37_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A37</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:37_1_1_2_1_3"/><span class="lab"><font color="Green" title="E57">A38</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <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#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E7:37_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A39</font></span>: 
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_2_1_3"><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="E8:37_1_1_2_1_3"/>
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E9:37_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A40</font></span>: 
<font color="Maroon" title="c36">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="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="txt" href="#E6:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:37_1_1_2_1_3"/>
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E2:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:37_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A41</font></span>: 
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1_1_2_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="txt" href="#E4:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E55">A36</font></span></a></span>;<br/><a NAME="E12:37_1_1_2_1_3"/>
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:37_1_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1_1_2_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E2:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="txt" href="#E5:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E56">A37</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:37_1_1_2_1_3"/>
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E2:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="txt" href="#E11:37_1_1_2_1_3"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E12:37_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A42</font></span>: 
<font color="Maroon" title="c43">h</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c36">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></span>;<br/><a NAME="E13:37_1_1_2_1"/><span class="lab"><font color="Green" title="E53">A43</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><a NAME="E14:37_1_1_2_1"/><span class="lab"><font color="Green" title="E54">A44</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E15:37_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A45</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:37_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A46</font></span>: 
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E13:37_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a></span>;<br/><a NAME="E17:37_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A47</font></span>: 
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E13:37_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E14:37_1_1_2_1"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>;<br/><a NAME="E18:37_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A48</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p5">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p5">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a></span>;<br/><a NAME="E19:37_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A49</font></span>: 
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E13:37_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E20:37_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E16:37_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E15:37_1_1_2_1"><span class="lab"><font color="Green" title="E55">A45</font></span></a>, <a class="txt" href="#E13:37_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>;<br/><a NAME="E21:37_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A50</font></span>: 
<font color="Maroon" title="c38">f1</font>,<font color="Maroon" title="c43">h</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E10:37_1_1_2_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E14:37_1_1_2_1"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="txt" href="#E15:37_1_1_2_1"><span class="lab"><font color="Green" title="E55">A45</font></span></a>, <a class="txt" href="#E13:37_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E17:37_1_1_2_1"><span class="lab"><font color="Green" title="E57">A47</font></span></a>, <a class="ref" href="scpisort.html#T4" target="_self" title="SCPISORT:th.4">Th4</a></span>;<br/><a NAME="E22:37_1_1_2_1"/><span class="lab"><font color="Green" title="E61">A51</font></span>: 
<font color="Maroon" title="c43">h</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font>,<font color="Maroon" title="c34">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1_2_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>;<br/><a NAME="E23:37_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">h</font>,<font color="Maroon" title="c39">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E17:37_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E18:37_1_1_2_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E12:37_1_1_2_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:37_1_1_2_1"/>
<font color="Maroon" title="c38">f1</font>,<font color="Maroon" title="c39">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:37_1_1_2_1"><span class="lab"><font color="Green" title="E60">A50</font></span></a>, <a class="ref" href="classes1.html#T84" title="CLASSES1:th.84">CLASSES1:84</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c39">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> holds <br/><font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/><a NAME="E25:37_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A52</font></span>: 
<font color="Maroon" title="c39">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E17:37_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E18:37_1_1_2_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E22:37_1_1_2_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E12:37_1_1_2_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c44">i</font>, <font color="Maroon" title="c45">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font> &amp; <font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1_2_1_4"/><span class="lab"><font color="Green" title="E63">A53</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_1_2_1_4"/><span class="lab"><font color="Green" title="E64">A54</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">and </span><br/><a NAME="E3:37_1_1_2_1_4"/><span class="lab"><font color="Green" title="E65">A55</font></span>: 
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_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="E1:37_1_1_2_1_4_1"/>
( <font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c45">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_2_1_4"><span class="lab"><font color="Green" title="E65">A55</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_4_1_1"/>
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_1_1_2_1_4_1_1"/>
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:37_1_1_2_1"><span class="lab"><font color="Green" title="E62">A52</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_4"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="txt" href="#E2:37_1_1_2_1_4"><span class="lab"><font color="Green" title="E64">A54</font></span></a>, <a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_4_1_2"/><span class="lab"><font color="Green" title="E66">A56</font></span>: 
<font color="Maroon" title="c45">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_1_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="E1:37_1_1_2_1_4_1_2_1_1"/>
( <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">j</font> or <font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c45">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_4_1_2_1_1_1"/>
<font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_1_1_2_1_4_1_2_1_1_1"/>
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_4_1_2_1_1_2"/>
<font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c45">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font></span><br/><div class="add"><a NAME="E2:37_1_1_2_1_4_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1_2_1_4"><span class="lab"><font color="Green" title="E64">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:37_1_1_2_1_4_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E66">A56</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c46">mm</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:37_1_1_2_1_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E67">A57</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">mm</font>
 <span class="kw">and </span><br/><a NAME="E6:37_1_1_2_1_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E68">A58</font></span>: 
<font color="Maroon" title="c46">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E7:37_1_1_2_1_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E69">A59</font></span>: 
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E17:37_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E18:37_1_1_2_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E22:37_1_1_2_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E12:37_1_1_2_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_4"><span class="lab"><font color="Green" title="E63">A53</font></span></a></span>;<br/><a NAME="E8:37_1_1_2_1_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A60</font></span>: 
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E17:37_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E18:37_1_1_2_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E22:37_1_1_2_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E12:37_1_1_2_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="txt" href="#E4:37_1_1_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E13:37_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E66">A56</font></span></a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_1_2_1_1_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="E1:37_1_1_2_1_4_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c46">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c46">mm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_4_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c46">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_1_1_2_1_4_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E13:37_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E15:37_1_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E16:37_1_1_2_1"><span class="lab"><font color="Green" title="E56">A46</font></span></a>, <a class="txt" href="#E5:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A57</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A59</font></span></a>, <a class="txt" href="#E8:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A60</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_4_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_4_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A61</font></span>: 
<font color="Maroon" title="c46">mm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font></span><br/><div class="add"><a NAME="E2:37_1_1_2_1_4_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c46">mm</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:37_1_1_2_1_4_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c46">mm</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c43">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E13:37_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:37_1_1_2_1_4_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A62</font></span>: 
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E10:37_1_1_2_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E4:37_1_1_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E5:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A57</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A58</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_4_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A61</font></span></a></span>;<br/><a NAME="E5:37_1_1_2_1_4_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E13:37_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E14:37_1_1_2_1"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="txt" href="#E16:37_1_1_2_1"><span class="lab"><font color="Green" title="E56">A46</font></span></a>, <a class="txt" href="#E8:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A60</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:37_1_1_2_1_4_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E5:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A57</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A58</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A59</font></span></a>, <a class="txt" href="#E4:37_1_1_2_1_4_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A62</font></span></a>, <a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E27:37_1_1_2_1"/>
<font color="Maroon" title="c39">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> holds <br/><font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> implies <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1_2_1_5"/><span class="lab"><font color="Green" title="E63">A63</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:37_1_1_2_1_5"/><span class="lab"><font color="Green" title="E64">A64</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font></span><br/><a NAME="E3:37_1_1_2_1_5"/><span class="lab"><font color="Green" title="E65">A65</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E4:37_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A66</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_2_1_5"><span class="lab"><font color="Green" title="E63">A63</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:37_1_1_2_1_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E6:37_1_1_2_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A67</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1_2_1_5"><span class="lab"><font color="Green" title="E66">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> = 
<font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E17:37_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E18:37_1_1_2_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E22:37_1_1_2_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E12:37_1_1_2_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="txt" href="#E2:37_1_1_2_1_5"><span class="lab"><font color="Green" title="E64">A64</font></span></a>, <a class="txt" href="#E4:37_1_1_2_1_5"><span class="lab"><font color="Green" title="E66">A66</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1_1_2_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_5"><span class="lab"><font color="Green" title="E63">A63</font></span></a>, <a class="txt" href="#E2:37_1_1_2_1_5"><span class="lab"><font color="Green" title="E64">A64</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1_5"><span class="lab"><font color="Green" title="E65">A65</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_5"><span class="lab"><font color="Green" title="E67">A67</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  ex <font color="Olive" title="b1">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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1_2_1_6"/><span class="lab"><font color="Green" title="E63">A68</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_1_2_1_6"/><span class="lab"><font color="Green" title="E64">A69</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_6_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="E1:37_1_1_2_1_6_1"/>
( <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_6_1_1"/><span class="lab"><font color="Green" title="E65">A70</font></span>: 
<font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c45">j</font> = <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font> &amp; <font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> )</span><br/><span class="kw">thus </span><a NAME="E2:37_1_1_2_1_6_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> )</span><br/><span class="kw">thus </span><a NAME="E3:37_1_1_2_1_6_1_1"/>
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font></span><br/><span class="kw">thus </span><a NAME="E4:37_1_1_2_1_6_1_1"/>
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E17:37_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E18:37_1_1_2_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E22:37_1_1_2_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E12:37_1_1_2_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="txt" href="#E4:37_1_1_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E13:37_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E17:37_1_1_2_1"><span class="lab"><font color="Green" title="E57">A47</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_6_1_1"><span class="lab"><font color="Green" title="E65">A70</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_6_1_2"/>
<font color="Maroon" title="c44">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><div class="add"><a NAME="E2:37_1_1_2_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="txt" href="#E2:37_1_1_2_1_6"><span class="lab"><font color="Green" title="E64">A69</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:37_1_1_2_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c45">mm</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:37_1_1_2_1_6_1_2"/><span class="lab"><font color="Green" title="E65">A71</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">mm</font>
 <span class="kw">and </span><br/><a NAME="E6:37_1_1_2_1_6_1_2"/><span class="lab"><font color="Green" title="E66">A72</font></span>: 
<font color="Maroon" title="c45">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E7:37_1_1_2_1_6_1_2"/><span class="lab"><font color="Green" title="E67">A73</font></span>: 
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E17:37_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E18:37_1_1_2_1"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E22:37_1_1_2_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E3:37_1_1_2_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E9:37_1_1_2_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E12:37_1_1_2_1"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_6"><span class="lab"><font color="Green" title="E63">A68</font></span></a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><a NAME="E1:37_1_1_2_1_6_1_2_1"/><span class="lab"><font color="Green" title="E68">A74</font></span>: 
<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_6_1_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="E1:37_1_1_2_1_6_1_2_1_1"/>
( <font color="Maroon" title="c45">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c45">mm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_6_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_6_1_2_1_1_1"/><span class="lab"><font color="Green" title="E69">A75</font></span>: 
<font color="Maroon" title="c45">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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 <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c46">j</font> = <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">j</font> &amp; <font color="Maroon" title="c46">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">j</font> )</span><br/><span class="kw">thus </span><a NAME="E2:37_1_1_2_1_6_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_2_1_6_1_2_1"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c46">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">j</font> )</span><br/><span class="kw">thus </span><a NAME="E3:37_1_1_2_1_6_1_2_1_1_1"/>
<font color="Maroon" title="c46">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">j</font></span><br/><span class="kw">thus </span><a NAME="E4:37_1_1_2_1_6_1_2_1_1_1"/>
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E16:37_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E15:37_1_1_2_1"><span class="lab"><font color="Green" title="E55">A45</font></span></a>, <a class="txt" href="#E13:37_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E19:37_1_1_2_1"><span class="lab"><font color="Green" title="E59">A49</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1_6_1_2"><span class="lab"><font color="Green" title="E67">A73</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_6_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A75</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_6_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_1_6_1_2_1_1_2"/><span class="lab"><font color="Green" title="E69">A76</font></span>: 
<font color="Maroon" title="c45">mm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">mm</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">mm</font> &amp; <font color="Olive" title="b1">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">mm</font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c46">mm</font> = <font color="Maroon" title="c45">mm</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">mm</font> &amp; <font color="Maroon" title="c46">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">mm</font> )</span><br/><span class="kw">thus </span><a NAME="E2:37_1_1_2_1_6_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1_2_1_6_1_2"><span class="lab"><font color="Green" title="E65">A71</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c46">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">mm</font> )</span><br/><span class="kw">thus </span><a NAME="E3:37_1_1_2_1_6_1_2_1_1_2"/>
<font color="Maroon" title="c46">mm</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="txt" href="#E4:37_1_1_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_6_1_2"><span class="lab"><font color="Green" title="E66">A72</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">mm</font></span><br/><a NAME="E4:37_1_1_2_1_6_1_2_1_1_2"/>
<font color="Maroon" title="c46">mm</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_6_1_2"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:37_1_1_2_1_6_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c46">mm</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37_1_1_2_1"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:37_1_1_2_1_6_1_2_1_1_2"/>
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1_1_2_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E4:37_1_1_2_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E5:37_1_1_2_1_6_1_2"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="txt" href="#E6:37_1_1_2_1_6_1_2"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="txt" href="#E7:37_1_1_2_1_6_1_2"><span class="lab"><font color="Green" title="E67">A73</font></span></a>, <a class="txt" href="#E1:37_1_1_2_1_6_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A76</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_2"/><span class="lab"><font color="Green" title="E45">A77</font></span>: 
<font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>5</sub></font> ) ) )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E46">A78</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c43">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> implies <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1_2_2_1"/><span class="lab"><font color="Green" title="E47">A79</font></span>: 
<font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:37_1_1_2_2_1"/><span class="lab"><font color="Green" title="E48">A80</font></span>: 
<font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E3:37_1_1_2_2_1"/><span class="lab"><font color="Green" title="E49">A81</font></span>: 
<font color="Maroon" title="c43">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E4:37_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A82</font></span>: 
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="txt" href="#E2:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">A80</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>;<br/><a NAME="E5:37_1_1_2_2_1"/><span class="lab"><font color="Green" title="E51">A83</font></span>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p5">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> </span>)</span>,6</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E15:37_1_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2"><span class="lab"><font color="Green" title="E45">A77</font></span></a>, <a class="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></span>;<br/><a NAME="E6:37_1_1_2_2_1"/>
<font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E7:37_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A84</font></span>: 
<font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_1_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_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="E1:37_1_1_2_2_1_1"/>
( <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> or ( <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E53">A85</font></span>: 
<font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E13:37_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E15:37_1_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2"><span class="lab"><font color="Green" title="E45">A77</font></span></a>, <a class="txt" href="#E4:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E50">A82</font></span></a>, <a class="txt" href="#E5:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E51">A83</font></span></a>, <a class="ref" href="scmpds_8.html#T23" title="SCMPDS_8:th.23">SCMPDS_8:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="scpisort.html#E33"><span class="lab"><font color="Green" title="E23">Lm15</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E13:37_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E53">A85</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E53">A86</font></span>: 
<font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E16:37_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E15:37_1_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2"><span class="lab"><font color="Green" title="E45">A77</font></span></a>, <a class="txt" href="#E4:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E50">A82</font></span></a>, <a class="txt" href="#E5:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E51">A83</font></span></a>, <a class="ref" href="scmpds_8.html#T23" title="SCMPDS_8:th.23">SCMPDS_8:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="scpisort.html#E33"><span class="lab"><font color="Green" title="E23">Lm15</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E16:37_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="txt" href="#E2:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">A80</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E53">A86</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_2_1_1_3"/>
( <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E7:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E52">A84</font></span></a>, <a class="txt" href="#E4:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E50">A82</font></span></a>, <a class="txt" href="scpisort.html#E24"><span class="lab"><font color="Green" title="E16">Lm8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E51">A83</font></span></a>, <a class="ref" href="scmpds_8.html#T23" title="SCMPDS_8:th.23">SCMPDS_8:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c37">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="scpisort.html#E33"><span class="lab"><font color="Green" title="E23">Lm15</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E3:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E49">A81</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="txt" href="#E2:37_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">A80</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:37_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A87</font></span>: 
<font color="Maroon" title="c38">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><span class="kw">thus </span><a NAME="E4:37_1_1_2_2"/>
<font color="Maroon" title="c38">f1</font>,<font color="Maroon" title="c39">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E2:37_1_1_2_2"><span class="lab"><font color="Green" title="E46">A78</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c39">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> holds <br/><font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c43">i</font>, <font color="Maroon" title="c44">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">i</font> &amp; <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1_2_2_2"/><span class="lab"><font color="Green" title="E48">A88</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">i</font>
 <span class="kw">and </span><br/><a NAME="E2:37_1_1_2_2_2"/><span class="lab"><font color="Green" title="E49">A89</font></span>: 
<font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font>
 <span class="kw">and </span><br/><a NAME="E3:37_1_1_2_2_2"/><span class="lab"><font color="Green" title="E50">A90</font></span>: 
<font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_2_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="E1:37_1_1_2_2_2_1"/>
( <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_2_2_2"><span class="lab"><font color="Green" title="E50">A90</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_2_2_1_1"/>
<font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_1_1_2_2_2_1_1"/>
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_2"><span class="lab"><font color="Green" title="E48">A88</font></span></a>, <a class="txt" href="#E2:37_1_1_2_2_2"><span class="lab"><font color="Green" title="E49">A89</font></span></a>, <a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_2_2_1_2"/><span class="lab"><font color="Green" title="E51">A91</font></span>: 
<font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_2_1_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="E1:37_1_1_2_2_2_1_2_1_1"/>
( <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">j</font> or <font color="Maroon" title="c43">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_2_2_1_2_1_1_1"/>
<font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_1_1_2_2_2_1_2_1_1_1"/>
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_2_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_1_2_2_2_1_2_1_1_2"/>
<font color="Maroon" title="c43">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c44">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font></span><br/><div class="add"><a NAME="E2:37_1_1_2_2_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1_2_2_2"><span class="lab"><font color="Green" title="E49">A89</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:37_1_1_2_2_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_2_2_2_1_2"><span class="lab"><font color="Green" title="E51">A91</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:37_1_1_2_2_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A92</font></span>: 
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_2"><span class="lab"><font color="Green" title="E48">A88</font></span></a>, <a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></span>;<br/><a NAME="E5:37_1_1_2_2_2_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">k</font> <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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E6:37_1_1_2_2_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A93</font></span>: 
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E13:37_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>;<br/><a NAME="E7:37_1_1_2_2_2_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_2_1_2"><span class="lab"><font color="Green" title="E51">A91</font></span></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E8:37_1_1_2_2_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c37">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E16:37_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E1:37_1_1_2_2_2_1_2"><span class="lab"><font color="Green" title="E51">A91</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:37_1_1_2_2_2_1_2_1_1_2"/>
<font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_2_2"><span class="lab"><font color="Green" title="E45">A77</font></span></a>, <a class="txt" href="#E4:37_1_1_2_2_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E52">A92</font></span></a>, <a class="txt" href="#E6:37_1_1_2_2_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A93</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="Red">::  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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:37_1_1_2_2"/>
<font color="Maroon" title="c39">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="txt" href="#E3:37_1_1_2_2"><span class="lab"><font color="Green" title="E47">A87</font></span></a>, <a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> holds <br/><font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E7:37_1_1_2_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f1</font> holds <br/><font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E2:37_1_1_2_2"><span class="lab"><font color="Green" title="E46">A78</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> )</span><br/><span class="kw">thus </span><a NAME="E8:37_1_1_2_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_2_2"><span class="lab"><font color="Green" title="E47">A87</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:37_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c36">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:37"/><span class="lab"><font color="Green" title="E28">A94</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c36">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Olive" title="b1">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c36">t</font>,<font color="Maroon" title="c34">m</font> &amp; <font color="Olive" title="b2">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c36">t</font>,<font color="Maroon" title="c34">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f1</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>  &amp; <font color="Olive" title="b1">f1</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f1</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c37">f1</font>, <font color="Maroon" title="c38">f2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span> &amp; <font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> &amp; <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c37">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c36">t</font>,<font color="Maroon" title="c34">m</font> &amp; <font color="Maroon" title="c38">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c36">t</font>,<font color="Maroon" title="c34">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</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>  &amp; <font color="Maroon" title="c37">f1</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c37">f1</font>,<font color="Maroon" title="c38">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c38">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> holds <br/><font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_2"/>
<font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:37_2"/><span class="lab"><font color="Green" title="E29">A95</font></span>: 
( <font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span> )
 <span class="kw">and </span><br/><a NAME="E3:37_2"/>
<font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 6</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E4:37_2"/><span class="lab"><font color="Green" title="E30">A96</font></span>: 
<font color="Maroon" title="c37">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c36">t</font>,<font color="Maroon" title="c34">m</font>
 <span class="kw">and </span><br/><a NAME="E5:37_2"/><span class="lab"><font color="Green" title="E31">A97</font></span>: 
<font color="Maroon" title="c38">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p3">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c36">t</font>,<font color="Maroon" title="c34">m</font>
 <span class="kw">and </span><br/><a NAME="E6:37_2"/><span class="lab"><font color="Green" title="E32">A98</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">f2</font>
 <span class="kw">and </span><br/><a NAME="E7:37_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</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">and </span><br/><a NAME="E8:37_2"/>
<font color="Maroon" title="c37">f1</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c37">f1</font>,<font color="Maroon" title="c38">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c38">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> holds <br/><font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/>

<a NAME="E9:37_2"/><span class="lab"><font color="Green" title="E33">A99</font></span>: 
<font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> </span>)</span>,6</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_2"><span class="lab"><font color="Green" title="E29">A95</font></span></a>, <a class="ref" href="scmp_gcd.html#T5" title="SCMP_GCD:th.5">SCMP_GCD:5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E34">A100</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c39">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">i</font> &amp; <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> implies <font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:37_2_1"/><span class="lab"><font color="Green" title="E35">A101</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">i</font> &amp; <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font></span><br/><a NAME="E2:37_2_1"/><span class="lab"><font color="Green" title="E36">A102</font></span>: 
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> = 
<font color="Maroon" title="c36">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c39">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2"><span class="lab"><font color="Green" title="E30">A96</font></span></a>, <a class="txt" href="#E1:37_2_1"><span class="lab"><font color="Green" title="E35">A101</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K14" title="SCMPDS_2:func.14">SubFrom</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p4">(<span class="default"><a href="scmpds_6.html#K3" title="SCMPDS_6:func.3">if&gt;0</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5 <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">';'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <span class="p4">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> 4</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="scmpds_2.html#K17" title="SCMPDS_2:func.17">:=</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,5</span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,4,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p0">(<span class="default"><a href="scmpds_2.html#K12" title="SCMPDS_2:func.12">AddTo</a> <a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="scmpds_4.html#K1" title="SCMPDS_4:func.1">Load</a> <span class="p0">(<span class="default"><a href="scmp_gcd.html#K2" title="SCMP_GCD:func.2">GBP</a> ,6 <a href="scmpds_2.html#K11" title="SCMPDS_2:func.11">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c36">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c34">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c39">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_2"><span class="lab"><font color="Green" title="E33">A99</font></span></a>, <a class="ref" href="scmpds_8.html#T23" title="SCMPDS_8:th.23">SCMPDS_8:23</a></span>
<br/>.= 
<font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_2"><span class="lab"><font color="Green" title="E31">A97</font></span></a>, <a class="txt" href="#E6:37_2"><span class="lab"><font color="Green" title="E32">A98</font></span></a>, <a class="txt" href="#E2:37_2_1"><span class="lab"><font color="Green" title="E36">A102</font></span></a>, <a class="txt" href="#E1:37_2_1"><span class="lab"><font color="Green" title="E35">A101</font></span></a>, <a class="ref" href="scpisort.html#D1" target="_self" title="SCPISORT:def.1">Def1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:37_2"/>
<font color="Maroon" title="c37">f1</font>,<font color="Maroon" title="c38">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_2"><span class="lab"><font color="Green" title="E32">A98</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c38">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> holds <br/><font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:37_2"/>
<font color="Maroon" title="c38">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="scpisort.html#T1" target="_self" title="SCPISORT:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> holds <br/><font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:37_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">f1</font> holds <br/><font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_2"><span class="lab"><font color="Green" title="E34">A100</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> )</span><br/>

<a NAME="E14:37_2"/>
<font color="Maroon" title="c37">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_2"><span class="lab"><font color="Green" title="E32">A98</font></span></a>, <a class="txt" href="#E10:37_2"><span class="lab"><font color="Green" title="E34">A100</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:37_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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 href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c38">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:37"/><span class="lab"><font color="Green" title="E29">A103</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2: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="#E4:37"><span class="lab"><font color="Green" title="E28">A94</font></span></a>, <a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E27">A3</font></span></a>);<br/></span>
<span class="kw">assume </span><a NAME="E6:37"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> &amp; <font color="Maroon" title="c32">f</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c32">f</font>,<font color="Maroon" title="c33">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c33">g</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> holds <br/><font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )</span><br/>

<span class="kw">hence </span><a NAME="E7:37"/>
( <font color="Maroon" title="c32">f</font>,<font color="Maroon" title="c33">g</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c33">g</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">f</font> holds <br/><font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</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="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c33">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E29">A103</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
