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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">Bn</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b4">l</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Olive" title="b2">Bn</font>  st <font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">Bn</font> is  <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Olive" title="b4">l</font> &amp; <font color="Olive" title="b3">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Bn</font> &amp; <font color="Maroon" title="c2">j</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"><font color="Olive" title="b4">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Olive" title="b1">F</font></span>)</span> &amp; <font color="Olive" title="b3">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> holds <br/><span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">v0</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v0</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">Bn</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">l</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Olive" title="b1">Bn</font>  st <font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">Bn</font> is  <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Olive" title="b3">l</font> &amp; <font color="Olive" title="b2">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Bn</font> &amp; <font color="Maroon" title="c2">j</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"><font color="Olive" title="b3">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp; <font color="Olive" title="b2">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> holds <br/><span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">v0</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v0</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">Bn</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">v0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">l</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">Bn</font>  st <font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">Bn</font> is  <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Olive" title="b2">l</font> &amp; <font color="Olive" title="b1">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Bn</font> &amp; <font color="Maroon" title="c2">j</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"><font color="Olive" title="b2">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp; <font color="Olive" title="b1">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> holds <br/><span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">v0</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v0</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">v0</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">l</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">Bn</font>  st <font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">Bn</font> is  <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Olive" title="b1">l</font> &amp; <font color="Maroon" title="c5">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Bn</font> &amp; <font color="Maroon" title="c2">j</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"><font color="Olive" title="b1">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp; <font color="Maroon" title="c5">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> holds <br/><span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">l</font> be    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">Bn</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="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">Bn</font> is  <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c6">l</font> &amp; <font color="Maroon" title="c5">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Bn</font> &amp; <font color="Maroon" title="c2">j</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"><font color="Maroon" title="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp; <font color="Maroon" title="c5">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> implies <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E2:73"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c4">Bn</font> is  <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a> 
 <span class="kw">and </span><br/><a NAME="E3:73"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c6">l</font>
 <span class="kw">and </span><br/><a NAME="E4:73"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c5">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Bn</font>
 <span class="kw">and </span><br/><a NAME="E5:73"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c2">j</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"><font color="Maroon" title="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:73"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c5">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font>
 ; <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"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)</span><br/>

<a NAME="E7:73"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D7" title="RLVECT_2:def.7">RLVECT_2:def 7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E47">A8</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="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> = 
 <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="c3">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">F2</font> = <font color="Maroon" title="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">rv0</font> = <font color="Maroon" title="c5">v0</font> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/>
<a NAME="E11:73"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c6">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Bn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<a NAME="E12:73"/><span class="lab"><font color="Green" title="E49">A10</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="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E13:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:73"/><span class="lab"><font color="Green" title="E51">A12</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:73"/><span class="lab"><font color="Green" title="E52">A13</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:73"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
 for <font color="Olive" title="b1">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">j2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> implies <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>) );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> implies <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  );<br/>
<a NAME="E18:73"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>) = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:73"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="real_ns1.html#D5" title="REAL_NS1:def.5">REAL_NS1:def 5</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T18" title="EUCLID_4:th.18">EUCLID_4:18</a></span>
;<br/>
<a NAME="E20:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E21:73"/><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon" title="c2">j</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A7</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="E22:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E23:73"/><span class="lab"><font color="Green" title="E57">A18</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_3"/><span class="lab"><font color="Green" title="E58">A19</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</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">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_3_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="E1:73_3_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font> or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:73_3_1_1_1"/><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font>
 ; <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 class="add"><a NAME="E2:73_3_1_1_1"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:73_3_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_3_1_1_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:73_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_3_1_1_1"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:73_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:73_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><a NAME="E8:73_3_1_1_1"/><span class="lab"><font color="Green" title="E61">A22</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="txt" href="#E1:73_3_1_1_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">rv</font> = <font color="Maroon" title="c11">v</font> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">fk</font> = <font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_3_1_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="E1:73_3_1_1_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_3_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A23</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</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">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:73_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A24</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:73_3_1_1_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:73"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="txt" href="#E1:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:73_3_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A24</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:73_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A25</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:73_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A26</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <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="E7:73_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A27</font></span>: 
<font color="Maroon" title="c8">rv0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E8:73"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E1:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A23</font></span></a>, <a class="txt" href="#E5:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">fk1</font> = <font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/><a NAME="E9:73_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A28</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E10:73_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E68">A29</font></span>: 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">fk</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c12">rv</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c13">fk</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c12">rv</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T28" title="EUCLID_4:th.28">EUCLID_4:28</a></span>;<br/><a NAME="E11:73_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A30</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="txt" href="#E6:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E12:73_3_1_1_1_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E5:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="ref" href="rlvect_2.html#D7" title="RLVECT_2:def.7">RLVECT_2:def 7</a></span>;<br/><span class="kw">then </span><span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c12">rv</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c14">fk1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T22" title="EUCLID_4:th.22">EUCLID_4:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E11:73"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E11:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="txt" href="#E7:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="ref" href="euclid_7.html#D3" target="_self" title="EUCLID_7:def.3">Def3</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><a NAME="E14:73_3_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">fk</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c12">rv</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_3"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="txt" href="#E1:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A23</font></span></a>, <a class="txt" href="#E9:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A28</font></span></a>, <a class="txt" href="#E10:73_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="ref" href="real_ns1.html#D5" title="REAL_NS1:def.5">REAL_NS1:def 5</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E15:73_3_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_3_1_1_1"><span class="lab"><font color="Green" title="E61">A22</font></span></a>, <a class="ref" href="real_ns1.html#D5" title="REAL_NS1:def.5">REAL_NS1:def 5</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="73_3_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_3_1_1_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</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">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:73_3_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:73_3_1_1_2"/><span class="lab"><font color="Green" title="E59">A31</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</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">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:73_3_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E3:73_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_3_1_1_2"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:73_3_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:73"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E3:73_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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>
<a NAME="E24:73"/><span class="lab"><font color="Green" title="E58">A32</font></span>: 
 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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E20:73"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="txt" href="#E23:73"><span class="lab"><font color="Green" title="E57">A18</font></span></a>);<br/></span>
<a NAME="E25:73"/><span class="lab"><font color="Green" title="E59">A33</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> holds <br/><span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> implies <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">i0</font> = <font color="Maroon" title="c10">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:73_4"/>
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font>
 ; <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"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E3:73_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">i0</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:73"><span class="lab"><font color="Green" title="E58">A32</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:73_4"/>
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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>
<a NAME="E26:73"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_5"/><span class="lab"><font color="Green" title="E61">A35</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_5_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="E1:73_5_1"/>
( <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_5_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:73_5_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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="73_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_5_1_2"/><span class="lab"><font color="Green" title="E62">A36</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_5_1_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="E1:73_5_1_2_1"/>
( <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_5_1_2"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_5_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_5_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A37</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">j</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_5_1_2_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="E1:73_5_1_2_1_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font> or <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_5_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_5_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A38</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:73_5_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:73_5_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A39</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:73_5_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A40</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A7</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="E5:73_5_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A41</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <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="E6:73_5_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A42</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E7:73_5_1_2_1_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E65">A39</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="E8:73_5_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">fk1</font> = <font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">fk</font> = <font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/><a NAME="E12:73_5_1_2_1_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E13:73_5_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A43</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:73_5_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A44</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73"><span class="lab"><font color="Green" title="E53">A14</font></span></a></span>;<br/><a NAME="E15:73_5_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A45</font></span>: 
<font color="Maroon" title="c8">rv0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E8:73"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E1:73_5_1_2_1_1"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="txt" href="#E4:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E5:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">rv</font> = <font color="Maroon" title="c11">v</font> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/><a NAME="E17:73_5_1_2_1_1_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E4:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="rlvect_2.html#D7" title="RLVECT_2:def.7">RLVECT_2:def 7</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A46</font></span>: <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c14">rv</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c12">fk1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T22" title="EUCLID_4:th.22">EUCLID_4:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E11:73"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E6:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="txt" href="#E15:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="ref" href="euclid_7.html#D3" target="_self" title="EUCLID_7:def.3">Def3</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><a NAME="E19:73_5_1_2_1_1_1_1"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">fk</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c14">rv</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c13">fk</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c14">rv</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T28" title="EUCLID_4:th.28">EUCLID_4:28</a></span>;<br/><a NAME="E20:73_5_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">fk</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c14">rv</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_5"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="txt" href="#E1:73_5_1_2_1_1"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="txt" href="#E13:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E18:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="real_ns1.html#D5" title="REAL_NS1:def.5">REAL_NS1:def 5</a>, <a class="ref" href="rlvect_2.html#D7" title="RLVECT_2:def.7">RLVECT_2:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E21:73_5_1_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:73_5_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="real_ns1.html#D5" title="REAL_NS1:def.5">REAL_NS1:def 5</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="73_5_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_5_1_2_1_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:73_5_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D7" title="RLVECT_2:def.7">RLVECT_2:def 7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_5_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_5_1_2_1_2"/><span class="lab"><font color="Green" title="E63">A47</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:73_5_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">rv</font> = <font color="Maroon" title="c11">v</font> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/><a NAME="E5:73_5_1_2_1_2"/><span class="lab"><font color="Green" title="E64">A48</font></span>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E1:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="ref" href="rlvect_2.html#D7" title="RLVECT_2:def.7">RLVECT_2:def 7</a></span>;<br/><a NAME="E6:73_5_1_2_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E12:73"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E1:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A49</font></span>: <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c12">rv</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">rv0</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E1:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="txt" href="#E5:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E64">A48</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D5" title="REAL_NS1:def.5">REAL_NS1:def 5</a></span>
;<br/><a NAME="E8:73_5_1_2_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E9:73_5_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E13:73"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="txt" href="#E1:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:73_5_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A50</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73"><span class="lab"><font color="Green" title="E53">A14</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">fk</font> = <font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font> as    <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">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D6" title="REAL_NS1:def.6">REAL_NS1:def 6</a></span>;<br/><a NAME="E12:73_5_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">k</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:73"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="txt" href="#E1:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E13:73_5_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A51</font></span>: 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c13">fk</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D5" title="REAL_NS1:def.5">REAL_NS1:def 5</a></span>;<br/><a NAME="E14:73_5_1_2_1_2"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">fk</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c12">rv</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c13">fk</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c8">rv0</font>,<font color="Maroon" title="c12">rv</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T28" title="EUCLID_4:th.28">EUCLID_4:28</a></span>;<br/><span class="kw">hence </span><a NAME="E15:73_5_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="txt" href="#E13:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E67">A51</font></span></a>, <a class="txt" href="#E7:73_5_1_2_1_2"><span class="lab"><font color="Green" title="E65">A49</font></span></a>, <a class="ref" href="real_ns1.html#D5" title="REAL_NS1:def.5">REAL_NS1:def 5</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E27:73"/><span class="lab"><font color="Green" title="E61">A52</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:73"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E28:73"/><span class="lab"><font color="Green" title="E62">A53</font></span>: 
 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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E27:73"><span class="lab"><font color="Green" title="E61">A52</font></span></a>, <a class="txt" href="#E26:73"><span class="lab"><font color="Green" title="E60">A34</font></span></a>);<br/></span>
<a NAME="E29:73"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> holds <br/><span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> implies <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73_6"/><span class="lab"><font color="Green" title="E63">A54</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font>
 <span class="kw">and </span><br/><a NAME="E2:73_6"/><span class="lab"><font color="Green" title="E64">A55</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font>
 ; <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"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">i0</font> = <font color="Maroon" title="c10">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:73_6"/>
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">i0</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:73"><span class="lab"><font color="Green" title="E62">A53</font></span></a>, <a class="txt" href="#E1:73_6"><span class="lab"><font color="Green" title="E63">A54</font></span></a>, <a class="txt" href="#E2:73_6"><span class="lab"><font color="Green" title="E64">A55</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:73_6"/>
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:73"/>
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">v0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v0</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:73"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E13:73"><span class="lab"><font color="Green" title="E50">A11</font></span></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>
