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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font>, <font color="Olive" title="b4">w</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</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="Olive" title="b2">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">h</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="Olive" title="b4">w</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">h</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Olive" title="b4">w</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</font>, <font color="Maroon" title="c5">w</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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="c3">g</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h</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="c5">w</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c5">w</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:5"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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="c3">g</font>
 <span class="kw">and </span><br/><a NAME="E2:5"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h</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="c5">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c5">w</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">KK</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c4">h</font>, <font color="Maroon" title="c8">W</font> = <font color="Maroon" title="c5">w</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c3">g</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">FH</font> = <font color="Maroon" title="c9">F</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <font color="Maroon" title="c7">H</font>, <font color="Maroon" title="c12">GW</font> = <font color="Maroon" title="c10">G</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <font color="Maroon" title="c8">W</font>, <font color="Maroon" title="c13">Th36W</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c8">W</font></span>)</span> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">FH</font> = <font color="Maroon" title="c11">FH</font>, <font color="Maroon" title="c15">GW</font> = <font color="Maroon" title="c12">GW</font>, <font color="Maroon" title="c16">Th36W</font> = <font color="Maroon" title="c13">Th36W</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T151" title="FINSEQ_2:th.151">FINSEQ_2:151</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">i</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">FH</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c15">GW</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Th36W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:5_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FH</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c15">GW</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Th36W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span><br/><a NAME="E2:5_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c17">i</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> <font color="Maroon" title="c14">FH</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_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="E1:5_1_1_1"/>
( <font color="Maroon" title="c17">i</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> <font color="Maroon" title="c2">f</font> or  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:5_1_1_1_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c17">i</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> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FH</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c15">GW</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Th36W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span><br/><div class="add"><a NAME="E2:5_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E3:5_1_1_1_1"/><span class="lab"><font color="Green" title="E12">A7</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="c9">F</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><a NAME="E4:5_1_1_1_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:5_1_1_1_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><a NAME="E6:5_1_1_1_1"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><a NAME="E7:5_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E5:5_1_1_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">fi</font> = <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>, <font color="Maroon" title="c19">gi</font> = <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_1_1_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E2:5_1_1_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a></span>;<br/><a NAME="E9:5_1_1_1_1"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c14">FH</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">fi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E10:5_1_1_1_1"/>
<font color="Maroon" title="c15">GW</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">gi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E5:5_1_1_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E6:5_1_1_1_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c14">FH</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c15">GW</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> = 
<font color="Maroon" title="c18">fi</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c19">gi</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E9:5_1_1_1_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="fvsum_1.html#T22" title="FVSUM_1:th.22">FVSUM_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E5:5_1_1_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="fvsum_1.html#T22" title="FVSUM_1:th.22">FVSUM_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c16">Th36W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E5:5_1_1_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E3:5_1_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
;<br/> <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="5_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:5_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">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> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FH</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c15">GW</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Th36W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:5_1_1_1_2"/><span class="lab"><font color="Green" title="E10">A12</font></span>: 
<font color="Maroon" title="c18">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> <font color="Maroon" title="c4">h</font>
 <span class="kw">and </span><br/><a NAME="E4:5_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A13</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">n</font>
 ;<br/><a NAME="E5:5_1_1_1_2"/><span class="lab"><font color="Green" title="E12">A14</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1_1_2"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E6:5_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A15</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E7:5_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A16</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><a NAME="E8:5_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A17</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><a NAME="E9:5_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1_1_2"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E7:5_1_1_1_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">hn</font> = <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c20">wn</font> = <font color="Maroon" title="c5">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_1_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E6:5_1_1_1_2"><span class="lab"><font color="Green" title="E13">A15</font></span></a></span>;<br/><a NAME="E11:5_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A18</font></span>: 
<font color="Maroon" title="c14">FH</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">hn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1_1_2"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E4:5_1_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E12:5_1_1_1_2"/><span class="lab"><font color="Green" title="E17">A19</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="c7">H</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c8">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">h</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c10">G</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="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a>, <a class="ref" href="finseq_2.html#T144" title="FINSEQ_2:th.144">FINSEQ_2:144</a></span>;<br/><a NAME="E13:5_1_1_1_2"/>
<font color="Maroon" title="c15">GW</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">wn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:5_1_1_1_2"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E4:5_1_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E7:5_1_1_1_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E8:5_1_1_1_2"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c14">FH</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c15">GW</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> = 
<font color="Maroon" title="c19">hn</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">wn</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E11:5_1_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="ref" href="fvsum_1.html#T22" title="FVSUM_1:th.22">FVSUM_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c8">W</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1_1_2"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E7:5_1_1_1_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="ref" href="fvsum_1.html#T22" title="FVSUM_1:th.22">FVSUM_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c16">Th36W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1_1_2"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E4:5_1_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E7:5_1_1_1_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E12:5_1_1_1_2"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
;<br/> <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="E4:5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">FH</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c15">GW</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Th36W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 ; <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>
<span class="kw">hence </span><a NAME="E8:5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c5">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T139" title="FINSEQ_2:th.139">FINSEQ_2:139</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/>


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