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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">k</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> implies ( <font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:11"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">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="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_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="E1:11_1_1"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:11_1_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</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="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><div class="add"><a NAME="E2:11_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>;<br/><a NAME="E3:11_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xxreal_3.html#T82" title="XXREAL_3:th.82">XXREAL_3:82</a></span>;<br/><span class="kw">hence </span><a NAME="E4:11_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</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="11_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:11_1_1_2"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</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="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><div class="add"><a NAME="E2:11_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>;<br/><a NAME="E3:11_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xxreal_3.html#T82" title="XXREAL_3:th.82">XXREAL_3:82</a></span>;<br/><span class="kw">hence </span><a NAME="E4:11_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1_2"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</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="E3:11"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></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="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_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="E1:11_2"/>
(  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1"><span class="kw">suppose </span></a><a NAME="E1:11_2_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</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="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><div class="add"><a NAME="E2:11_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>;<br/><a NAME="E3:11_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xxreal_3.html#T82" title="XXREAL_3:th.82">XXREAL_3:82</a></span>;<br/><span class="kw">hence </span><a NAME="E4:11_2_1"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_2_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</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="11_2_2"><span class="kw">suppose </span></a><a NAME="E1:11_2_2"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</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="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><div class="add"><a NAME="E2:11_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></span>;<br/><a NAME="E3:11_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xxreal_3.html#T82" title="XXREAL_3:th.82">XXREAL_3:82</a></span>;<br/><span class="kw">hence </span><a NAME="E4:11_2_2"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_2_2"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</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>

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