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<span class="kw">let </span><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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"> ( <font color="Maroon" title="c1">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">V99</a>() &amp; <font color="Maroon" title="c1">seq1</font> is  <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>  &amp; <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">V99</a>() &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>  implies ( <font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">V99</a>() &amp;  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c2">seq2</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
( <font color="Maroon" title="c1">seq1</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">V99</a>() &amp; <font color="Maroon" title="c1">seq1</font> is  <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>  )
 <span class="kw">and </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
( <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">V99</a>() &amp; <font color="Maroon" title="c2">seq2</font> is  <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</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"> ( <font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">V99</a>() &amp;  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c2">seq2</font></span>)</span> )</span><br/>

<a NAME="E3:24"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c2">seq2</font></span>)</span></span><br/>

<a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp; <font color="Maroon" title="c1">seq1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c1">seq1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="rinfsup1.html#T8" target="_self" onmouseover="rs('rinfsup1/T8')" onmouseout="rh()">Th8</a>, <a class="ref" href="seq_1.html#T8" onmouseover="rs('seq_1/T8')" onmouseout="rh()">SEQ_1:8</a></span>;<br/>
<a NAME="E2:24_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c2">seq2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T18" target="_self" onmouseover="rs('rinfsup1/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E3:24_1"/>
( <font color="Maroon" title="c2">seq2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c2">seq2</font> &amp;  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c1">seq1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="rinfsup1.html#T8" target="_self" onmouseover="rs('rinfsup1/T8')" onmouseout="rh()">Th8</a>, <a class="ref" href="rinfsup1.html#T18" target="_self" onmouseover="rs('rinfsup1/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c2">seq2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="xreal_1.html#T66" onmouseover="rs('xreal_1/T66')" onmouseout="rh()">XREAL_1:66</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:24"/>
( <font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">V99</a>() &amp;  <a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c1">seq1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K2" title="RINFSUP1:func.2">lower_bound</a> <font color="Maroon" title="c2">seq2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup1.html#T10" target="_self" onmouseover="rs('rinfsup1/T10')" onmouseout="rh()">Th10</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/>


</div>
