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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</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" 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">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/>





<a NAME="E1:7"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E2:7"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E3:7"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E4:7"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</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="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:7_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</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="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>
; <a class="txt" 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="E6:7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
