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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">s</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#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Olive" title="b1">n</font></span>)</span>  st (  for <font color="Olive" title="b3">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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">K</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b2">s</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">s</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#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span>  st (  for <font color="Olive" title="b2">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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">K</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">s</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</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#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">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">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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">K</font> ) implies <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">K</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E13">A1</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">K</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"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">K</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c4">m</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">reconsider </span><font color="Maroon" title="c6">s0</font> = <font color="Maroon" title="c3">s</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="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">K</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">K</font></span><br/><a NAME="E2:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>;<br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">s</font> = 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">s0</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D4" title="PDIFF_1:def.4">PDIFF_1:def 4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:17_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="pdiff_8.html#T2" target="_self" title="PDIFF_8:th.2">Th2</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>
<a NAME="E7:17"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">s0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="pdiff_8.html#T15" target="_self" title="PDIFF_8:th.15">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:17"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T1" title="REAL_NS1:th.1">REAL_NS1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
