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<div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">fZ</font> = <font color="Maroon" title="c6">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" onmouseover="rs('partfun1/T45')" onmouseout="rh()">PARTFUN1:45</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">W</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vsdiff_1.html#K6" title="VSDIFF_1:func.6">bD</a> (<font color="Olive" title="b1">g</font>,<font color="Maroon" title="c4">h</font>) );<br/>
<a NAME="E2:26_1_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>)  ex <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>)  ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>); <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">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c10">x9</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T46" onmouseover="rs('partfun1/T46')" onmouseout="rh()">PARTFUN1:46</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> =  <a href="vsdiff_1.html#K6" title="VSDIFF_1:func.6">bD</a> (<font color="Maroon" title="c10">x9</font>,<font color="Maroon" title="c4">h</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" onmouseover="rs('partfun1/T45')" onmouseout="rh()">PARTFUN1:45</a></span>;<br/>
<a NAME="E3:26_1_1_1"/>
 ex <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">W</font> st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vsdiff_1.html#K6" title="VSDIFF_1:func.6">bD</a> (<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c4">h</font>) )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:26_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
 ; <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">consider </span><font color="Maroon" title="c8">g</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:26_1_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
( <font color="Maroon" title="c8">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fZ</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" onmouseover="rs('recdef_1/S2')" onmouseout="rh()">RECDEF_1:sch 2</a>(<a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">g</font>
; <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="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">f</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="vsdiff_1.html#K6" title="VSDIFF_1:func.6">bD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c4">h</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:26_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="vsdiff_1.html#K6" title="VSDIFF_1:func.6">bD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c4">h</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">i</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"> <font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="vsdiff_1.html#K6" title="VSDIFF_1:func.6">bD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<font color="Maroon" title="c4">h</font>)</span><br/>

<a NAME="E7:26_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:26_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="vsdiff_1.html#K6" title="VSDIFF_1:func.6">bD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<font color="Maroon" title="c4">h</font>)
 ; <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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