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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b2">r</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>; <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">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">r</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">r</font></span>)</span> )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">x</font> as  <font color="Maroon" title="c1">m</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">q</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> as  <font color="Maroon" title="c1">m</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c3">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font></span>)</span> as  <font color="Maroon" title="c1">m</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">t</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">r</font></span>)</span> as  <font color="Maroon" title="c1">m</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E5:15"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</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> <font color="Maroon" title="c1">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T89" title="FINSEQ_1:th.89">FINSEQ_1:89</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c3">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E7:15"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D5" title="PDIFF_1:def.5">PDIFF_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y1</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E9:15"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_7.html#T5" title="FINSEQ_7:th.5">FINSEQ_7:5</a></span>;<br/>
<a NAME="E11:15"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c5">p</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">k</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="c11">k</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="c5">p</font> implies <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c11">k</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="c5">p</font>
 ; <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="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/>

<a NAME="E2:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A6</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E3:15_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">k</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"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E5:15_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T27" title="RVSUM_1:th.27">RVSUM_1:27</a></span>;<br/>
<a NAME="E6:15_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D5" title="PDIFF_1:def.5">PDIFF_1:def 5</a></span>;<br/>
<a NAME="E7:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_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="E1:15_1_1"/>
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ; <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="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/><div class="add"><a NAME="E2:15_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> &amp; <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E5:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E7:15_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="finseq_7.html#T8" title="FINSEQ_7:th.8">FINSEQ_7:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:15_1_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ; <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="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/><div class="add"><a NAME="E2:15_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E7:15_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="finseq_7.html#T10" title="FINSEQ_7:th.10">FINSEQ_7:10</a></span>;<br/><a NAME="E3:15_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_2"/>
<font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E5:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:15"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</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"> <font color="Maroon" title="c3">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">r</font></span>)</span></span><br/>

<a NAME="E13:15"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c7">s</font> holds <br/><font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">k</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="c11">k</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="c7">s</font> implies <font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_2"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c11">k</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="c7">s</font>
 ; <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="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/>

<a NAME="E2:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A12</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E3:15_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">k</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"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E5:15_2"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T27" title="RVSUM_1:th.27">RVSUM_1:27</a></span>;<br/>
<a NAME="E6:15_2"/>
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">r</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D5" title="PDIFF_1:def.5">PDIFF_1:def 5</a></span>;<br/>
<a NAME="E7:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">r</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:15_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_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="E1:15_2_1"/>
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_2_1_1"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ; <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="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/><div class="add"><a NAME="E2:15_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E5:15_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="#E7:15_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="finseq_7.html#T8" title="FINSEQ_7:th.8">FINSEQ_7:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:15_2_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_2_1_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_2_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ; <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="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/><div class="add"><a NAME="E2:15_2_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E4:15_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E7:15_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="finseq_7.html#T10" title="FINSEQ_7:th.10">FINSEQ_7:10</a></span>;<br/><a NAME="E3:15_2_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:15_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_2_1_2"/>
<font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:15_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E5:15_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:15"/>
<font color="Maroon" title="c3">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c4">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E12">A1</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/>


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