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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> or ( not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><a NAME="E2:3_1_1_1"/><b>then </b>
( (  <a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or (  <a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i><a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/><a NAME="E3:3_1_1_1"/><b>then </b>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2"/>
( not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:3_1_1_2"/>
(  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_2"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:3"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="henmodel.html#K1">min*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


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