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<b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">h</font> be    <a href="prvect_1.html#M1" title="PRVECT_1:mode.1">BinOps</a> of  <a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> holds  <font color="Maroon" title="c2">f</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> holds  <font color="Maroon" title="c3">h</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) implies <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:54_1_2"/><i><font color="Green" title="E33">A2</font></i>: 
(  <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> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> holds  <font color="Maroon" title="c2">f</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:54_1_2"/><i><font color="Green" title="E34">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> holds  <font color="Maroon" title="c3">h</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">f'</font> = <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c5">h'</font> = <font color="Maroon" title="c3">h</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E4:54_1_2"/><i><font color="Green" title="E35">A4</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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="c4">f'</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">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="c5">h'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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="c4">f'</font> implies <font color="Maroon" title="c4">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><b>assume </b><a NAME="E1:54_1_2_1"/>
<font color="Maroon" title="c6">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="c4">f'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><b>then reconsider </b><font color="Maroon" title="c7">i'</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">g</font></span>)</span> <b>by </b><i><a class="txt" href="prvect_1.html#E1:54_1_2"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:54_1_2_1"/>
<font color="Maroon" title="c4">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c7">i'</font></span>)</span>
 <b>by </b><i><a class="txt" href="prvect_1.html#E1:54_1_2"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:54_1_2_1"/>
<font color="Maroon" title="c4">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="prvect_1.html#E2:54_1_2"><i><font color="Green" title="E34">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:54_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font>
 <b>by </b><i><a class="txt" href="prvect_1.html#E1:54_1_2"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="prvect_1.html#E2:54_1_2"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="prvect_1.html#E4:54_1_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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