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<span class="kw">let </span><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font> be   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">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="c2">f2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c2">f2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">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="c2">f2</font>
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c2">f2</font></span>)</span></span><br/>

<a NAME="E3:10"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <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="c1">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="rvsum_1.html#T115" onmouseover="rs('rvsum_1/T115')" onmouseout="rh()">RVSUM_1:115</a></span>;<br/>
<a NAME="E4:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="rfinseq2.html#D1" target="_self" onmouseover="rs('rfinseq2/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E5:10"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> &amp;  <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E7:10"/><span class="kw">then </span>
 <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="rfinseq2.html#D1" target="_self" onmouseover="rs('rfinseq2/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E9:10"/>
 <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="rfinseq2.html#D1" target="_self" onmouseover="rs('rfinseq2/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E11:10"/>
 <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:10"/>
 <a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" title="RFINSEQ2:func.3">max</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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