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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">Y</font> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> &amp; <font color="Olive" title="b1">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive" title="b1">f</font> <a href="nfcont_2.html#R1" title="NFCONT_2:pred.1">is_uniformly_continuous_on</a> <font color="Olive" title="b2">Y</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">Y</font> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> &amp; <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="nfcont_2.html#R1" title="NFCONT_2:pred.1">is_uniformly_continuous_on</a> <font color="Olive" title="b1">Y</font></span><br/><b>let </b><font color="Maroon" title="c4">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">Y</font> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> &amp; <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">Y</font> implies <font color="Maroon" title="c3">f</font> <a href="nfcont_2.html#R1" title="NFCONT_2:pred.1">is_uniformly_continuous_on</a> <font color="Maroon" title="c4">Y</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c4">Y</font> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="nfcont_2.html#R1" title="NFCONT_2:pred.1">is_uniformly_continuous_on</a> <font color="Maroon" title="c4">Y</font></span><br/>

<a NAME="E3:12"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="nfcont_1.html#D11" title="NFCONT_1:def.11">NFCONT_1:def 11</a></i>;<br/>
<b>assume </b><a NAME="E4:12"/>
not <font color="Maroon" title="c3">f</font> <a href="nfcont_2.html#R1" title="NFCONT_2:pred.1">is_uniformly_continuous_on</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c5">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:12"/><i><font color="Green" title="E11">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>and </b><br/><a NAME="E7:12"/><i><font color="Green" title="E12">A5</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b3">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> &amp; not <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><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="Olive" title="b2">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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="Olive" title="b3">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> )
 <b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="nfcont_2.html#D1" target="_self" title="NFCONT_2:def.1">Def1</a></i>;<br/>
<div><i><font color="Green" title="E13">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><a NAME="E1:12_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T5" title="NAT_1:th.5">NAT_1:5</a></i>;<br/><b>hence </b><a NAME="E2:12_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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 href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp;  ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default">$2 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b1">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; not <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $2</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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="Olive" title="b1">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> ) );<br/>
<div><i><font color="Green" title="E14">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Olive" title="b1">x1</font>]</span><br/><a NAME="E1:12_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E8:12"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green" title="E15">A8</font></i>: 
 ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b1">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; not <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><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="c7">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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="Olive" title="b1">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> )
 <b>by </b><i><a class="txt" href="#E7:12"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c8">x1</font>]</span><br/><b>thus </b><a NAME="E4:12_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c8">x1</font>]
 <b>by </b><i><a class="txt" href="#E3:12_2"><i><font color="Green" title="E15">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">s1</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E11:12"/><i><font color="Green" title="E15">A9</font></i>: 
for <font color="Olive" title="b1">n</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">n</font>,<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E9:12"><i><font color="Green" title="E14">A7</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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 href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> $2</span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; not <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $2</span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> );<br/>
<a NAME="E12:12"/><i><font color="Green" title="E16">A10</font></i>: 
for <font color="Olive" title="b1">n</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>   ex <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x2</font>]
 
<b>by </b><i><a class="txt" href="#E11:12"><i><font color="Green" title="E15">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">s2</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E14:12"/><i><font color="Green" title="E17">A11</font></i>: 
for <font color="Olive" title="b1">n</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">n</font>,<font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E12:12"><i><font color="Green" title="E16">A10</font></i></a>);<br/></i>
<div><i><font color="Green" title="E18">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font> implies  ex <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 <br/>for <font color="Olive" title="b2">m</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="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font> )</span><br/><b>assume </b><a NAME="E1:12_3"/><i><font color="Green" title="E19">A13</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <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 <br/>for <font color="Olive" title="b2">m</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="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font></span><br/><b>consider </b><font color="Maroon" title="c9">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> <b> such that </b><br/><a NAME="E3:12_3"/><i><font color="Green" title="E20">A14</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</a></i>;<br/><a NAME="E4:12_3"/><i><font color="Green" title="E21">A15</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="#E1:12_3"><i><font color="Green" title="E19">A13</font></i></a></i>;<br/><a NAME="E5:12_3"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:12_3"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E3:12_3"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:12_3"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E4:12_3"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="xreal_1.html#T78" title="XREAL_1:th.78">XREAL_1:78</a></i>;<br/><a NAME="E8:12_3"/><b>then </b><i><font color="Green" title="E22">A17</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T218" title="XCMPLX_1:th.218">XCMPLX_1:218</a></i>;<br/><b>take </b><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</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" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font></span><br/><b>let </b><font color="Maroon" title="c11">m</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> implies <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font> )</span><br/><b>assume </b><a NAME="E9:12_3"/><i><font color="Green" title="E23">A18</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font></span><br/><a NAME="E10:12_3"/>
<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> <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E9:12_3"><i><font color="Green" title="E23">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:12_3"/><b>then </b><i><font color="Green" title="E24">A19</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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>
 <b>by </b><i><a class="txt" href="#E3:12_3"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="#E4:12_3"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="xreal_1.html#T120" title="XREAL_1:th.120">XREAL_1:120</a></i>;<br/><i><font color="Green" title="E25">A20</font></i>: <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
;<br/><a NAME="E13:12_3"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E14:12"><i><font color="Green" title="E17">A11</font></i></a></i>;<br/><a NAME="E14:12_3"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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>
 <b>by </b><i><a class="txt" href="#E11:12_3"><i><font color="Green" title="E24">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E15:12_3"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="txt" href="#E8:12_3"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="#E12:12_3"><i><font color="Green" title="E25">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E16:12"/><b>then </b><i><font color="Green" title="E19">A21</font></i>: 
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D9" title="NORMSP_1:def.9">NORMSP_1:def 9</a></i>;<br/>
<a NAME="E17:12"/><b>then </b><i><font color="Green" title="E20">A22</font></i>: 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E15:12"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="normsp_1.html#D11" title="NORMSP_1:def.11">NORMSP_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )</span><br/><b>assume </b><a NAME="E1:12_4"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><a NAME="E2:12_4"/><b>then </b>
 ex <font color="Olive" title="b1">n</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" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6" title="NFCONT_1:th.6">NFCONT_1:6</a></i>;<br/><b>hence </b><a NAME="E3:12_4"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E11:12"><i><font color="Green" title="E15">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E19:12"/><b>then </b><i><font color="Green" title="E21">A23</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">q1</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E21:12"/><i><font color="Green" title="E22">A24</font></i>: 
( <font color="Maroon" title="c8">q1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c6">s1</font> &amp; <font color="Maroon" title="c8">q1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="nfcont_1.html#D4" title="NFCONT_1:def.4">NFCONT_1:def 4</a></i>;<br/>
<a NAME="E22:12"/><i><font color="Green" title="E23">A25</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">q1</font>
 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E21:12"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="nfcont_1.html#D11" title="NFCONT_1:def.11">NFCONT_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">Ns1</font> being  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E24:12"/><i><font color="Green" title="E24">A26</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">s1</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font>
 <b>by </b><i><a class="txt" href="#E21:12"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="valued_0.html#D17" title="VALUED_0:def.17">VALUED_0:def 17</a></i>;<br/>
<a NAME="E25:12"/><i><font color="Green" title="E25">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 
<b>by </b><i><a class="txt" href="#E16:12"><i><font color="Green" title="E19">A21</font></i></a>, <a class="ref" href="lopban_3.html#T11" title="LOPBAN_3:th.11">LOPBAN_3:11</a></i>;<br/>
<a NAME="E26:12"/><i><font color="Green" title="E26">A29</font></i>: 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E16:12"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="#E17:12"><i><font color="Green" title="E20">A22</font></i></a>, <a class="ref" href="lopban_3.html#T12" title="LOPBAN_3:th.12">LOPBAN_3:12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span>;<br/>
<a NAME="E27:12"/><i><font color="Green" title="E27">A30</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 
<b>by </b><i><a class="txt" href="#E21:12"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="#E25:12"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="normsp_1.html#T35" title="NORMSP_1:th.35">NORMSP_1:35</a></i>;<br/>
<i><font color="Green" title="E28">A31</font></i>:  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span>
<b>by </b><i><a class="txt" href="#E21:12"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="#E25:12"><i><font color="Green" title="E25">A28</font></i></a>, <a class="txt" href="#E26:12"><i><font color="Green" title="E26">A29</font></i></a>, <a class="ref" href="normsp_1.html#T43" title="NORMSP_1:th.43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">q1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E24:12"><i><font color="Green" title="E24">A26</font></i></a>, <a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E30:12"/><b>then </b><i><font color="Green" title="E29">A32</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">s2</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<a NAME="E31:12"/><b>then </b><i><font color="Green" title="E30">A33</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c7">s2</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">s2</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )</span><br/><b>assume </b><a NAME="E1:12_7"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><a NAME="E2:12_7"/><b>then </b>
 ex <font color="Olive" title="b1">n</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" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6" title="NFCONT_1:th.6">NFCONT_1:6</a></i>;<br/><b>hence </b><a NAME="E3:12_7"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E14:12"><i><font color="Green" title="E17">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E33:12"/><b>then </b><i><font color="Green" title="E31">A34</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E34:12"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font>
 
<b>by </b><i><a class="txt" href="#E21:12"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="valued_0.html#T21" title="VALUED_0:th.21">VALUED_0:21</a></i>;<br/>
<a NAME="E35:12"/><b>then </b><i><font color="Green" title="E32">A35</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="#E19:12"><i><font color="Green" title="E21">A23</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:12"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:12"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="#E35:12"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E38:12"/><b>then </b><i><font color="Green" title="E33">A36</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E39:12"/><b>then </b><i><font color="Green" title="E34">A37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E21:12"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="#E22:12"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="nfcont_1.html#D9" title="NFCONT_1:def.9">NFCONT_1:def 9</a></i>;<br/>
<a NAME="E40:12"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">s2</font>
 
<b>by </b><i><a class="txt" href="#E31:12"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="valued_0.html#T21" title="VALUED_0:th.21">VALUED_0:21</a></i>;<br/>
<a NAME="E41:12"/><b>then </b><i><font color="Green" title="E35">A38</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="#E33:12"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E42:12"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E43:12"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="#E41:12"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E44:12"/><b>then </b><i><font color="Green" title="E36">A39</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E45:12"/><b>then </b><i><font color="Green" title="E37">A40</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E22:12"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="#E27:12"><i><font color="Green" title="E27">A30</font></i></a>, <a class="txt" href="#E28:12"><i><font color="Green" title="E28">A31</font></i></a>, <a class="ref" href="nfcont_1.html#D9" title="NFCONT_1:def.9">NFCONT_1:def 9</a></i>;<br/>
<a NAME="E46:12"/><b>then </b><i><font color="Green" title="E38">A41</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span> is <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>
 
<b>by </b><i><a class="txt" href="#E39:12"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="normsp_1.html#T35" title="NORMSP_1:th.35">NORMSP_1:35</a></i>;<br/>
<i><font color="Green" title="E39">A42</font></i>:  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E39:12"><i><font color="Green" title="E34">A37</font></i></a>, <a class="txt" href="#E45:12"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="normsp_1.html#T43" title="NORMSP_1:th.43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c12">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> <b> such that </b><br/><a NAME="E2:12_9"/><i><font color="Green" title="E40">A43</font></i>: 
for <font color="Olive" title="b1">m</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" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="#E6:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E46:12"><i><font color="Green" title="E38">A41</font></i></a>, <a class="txt" href="#E47:12"><i><font color="Green" title="E39">A42</font></i></a>, <a class="ref" href="normsp_1.html#D11" title="NORMSP_1:def.11">NORMSP_1:def 11</a></i>;<br/><a NAME="E3:12_9"/><i><font color="Green" title="E41">A44</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="#E2:12_9"><i><font color="Green" title="E40">A43</font></i></a></i>;<br/><a NAME="E4:12_9"/><i><font color="Green" title="E42">A45</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">q1</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6" title="NFCONT_1:th.6">NFCONT_1:6</a></i>;<br/><a NAME="E5:12_9"/><i><font color="Green" title="E43">A46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6" title="NFCONT_1:th.6">NFCONT_1:6</a></i>;<br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="#E38:12"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="funct_2.html#T186" title="FUNCT_2:th.186">FUNCT_2:186</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="#E44:12"><i><font color="Green" title="E36">A39</font></i></a>, <a class="ref" href="funct_2.html#T186" title="FUNCT_2:th.186">FUNCT_2:186</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="#E38:12"><i><font color="Green" title="E33">A36</font></i></a>, <a class="txt" href="#E4:12_9"><i><font color="Green" title="E42">A45</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="#E44:12"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="#E5:12_9"><i><font color="Green" title="E43">A46</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c9">Ns1</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="#E24:12"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="#E30:12"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">Ns1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/><b>hence </b><a NAME="E7:12_9"/>
contradiction
 <b>by </b><i><a class="txt" href="#E14:12"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="#E3:12_9"><i><font color="Green" title="E41">A44</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E49:12"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
