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<span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="mazurulm.html#V3" title="MAZURULM:attr.3">midpoints-preserving</a>  implies <font color="Maroon" title="c4">f</font> is  <a href="mazurulm.html#V2" title="MAZURULM:attr.2">Affine</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="mazurulm.html#V1" title="MAZURULM:attr.1">isometric</a> 
 <span class="kw">and </span><br/><a NAME="E2:58_1_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="mazurulm.html#V3" title="MAZURULM:attr.3">midpoints-preserving</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> is  <a href="mazurulm.html#V2" title="MAZURULM:attr.2">Affine</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">E</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="mazurulm.html#D2" target="_self" title="MAZURULM:def.2">MAZURULM:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">E</font><br/>  for <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">t</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> &amp; <font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 implies <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E3:58_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> &amp; <font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span><br/>

<a NAME="E4:58_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">s</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E6:58_1_1"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 <span class="kw">and </span><br/><a NAME="E7:58_1_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c8">s</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E8:58_1_1"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mazurulm.html#T2" target="_self" title="MAZURULM:th.2">Th2</a>, <a class="ref" href="measure6.html#T64" title="MEASURE6:th.64">MEASURE6:64</a></span>;<br/>
<a NAME="E9:58_1_1"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">stb</font> = <font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">1t</font> = 1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">s1ta</font> = <span class="p1">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">s1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">za</font> = <span class="p1">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">zb</font> = <font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>;<br/>
<a NAME="E10:58_1_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="mazurulm.html#T24" target="_self" title="MAZURULM:th.24">Th24</a></span>;<br/>
<a NAME="E11:58_1_1"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>;<br/>
<a NAME="E12:58_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="normsp_1.html#T19" title="NORMSP_1:th.19">NORMSP_1:19</a></span>;<br/>
<a NAME="E13:58_1_1"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
 <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="mazurulm.html#T10" target="_self" title="MAZURULM:th.10">Th10</a></span>;<br/>
<a NAME="E14:58_1_1"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="seq_2.html#T10" title="SEQ_2:th.10">SEQ_2:10</a></span>;<br/>
<a NAME="E15:58_1_1"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E14:58_1_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="mazurulm.html#T7" target="_self" title="MAZURULM:th.7">Th7</a></span>;<br/>
<a NAME="E16:58_1_1"/>
 <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="mazurulm.html#T10" target="_self" title="MAZURULM:th.10">Th10</a></span>;<br/>
<a NAME="E17:58_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A13</font></span>: 
 <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</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="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</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">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E13:58_1_1"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E15:58_1_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="normsp_1.html#T25" title="NORMSP_1:th.25">NORMSP_1:25</a></span>;<br/>
<a NAME="E18:58_1_1"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
 <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E15:58_1_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="mazurulm.html#T10" target="_self" title="MAZURULM:th.10">Th10</a></span>;<br/>
<a NAME="E19:58_1_1"/><span class="lab"><font color="Green" title="E53">A15</font></span>: 
 <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="mazurulm.html#T10" target="_self" title="MAZURULM:th.10">Th10</a></span>;<br/>
<a NAME="E20:58_1_1"/><span class="lab"><font color="Green" title="E54">A16</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</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="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p5">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span><br/>

<span class="lab"><font color="Green" title="E55">A17</font></span>: <span class="p1">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> = 
1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T2" title="VALUED_1:th.2">VALUED_1:2</a></span>
<br/>.= 
1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
;<br/>
<a NAME="E2:58_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:58_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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> <span class="kw"> such that </span><br/><a NAME="E5:58_1_1_1"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c16">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="urysohn1.html#D2" title="URYSOHN1:def.2">URYSOHN1:def 2</a></span>;<br/>
<a NAME="E6:58_1_1_1"/><span class="lab"><font color="Green" title="E57">A19</font></span>: 
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c16">m</font> &amp; <font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c16">m</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="urysohn1.html#D1" title="URYSOHN1:def.1">URYSOHN1:def 1</a></span>;<br/>
<a NAME="E7:58_1_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</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="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p5">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font> = 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p5">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T108" title="FUNCT_2:th.108">FUNCT_2:108</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p5">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D2" title="NORMSP_1:def.2">NORMSP_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="ref" href="ndiff_1.html#D3" title="NDIFF_1:def.3">NDIFF_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D3" title="NDIFF_1:def.3">NDIFF_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:58_1_1_1"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="txt" href="mazurulm.html#E40"><span class="lab"><font color="Green" title="E38">Lm11</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="ref" href="ndiff_1.html#D3" title="NDIFF_1:def.3">NDIFF_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D3" title="NDIFF_1:def.3">NDIFF_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c15">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D2" title="NORMSP_1:def.2">NORMSP_1:def 2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E21:58_1_1"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p4">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</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="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font>
 
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>

<br/>.= 
 <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p5">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="ndiff_1.html#K2" title="NDIFF_1:func.2">*</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E10:58_1_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E21:58_1_1"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E12:58_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="txt" href="#E17:58_1_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="nfcont_1.html#T18" title="NFCONT_1:th.18">NFCONT_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></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">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:58_1_1"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E19:58_1_1"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E20:58_1_1"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="ref" href="normsp_1.html#T25" title="NORMSP_1:th.25">NORMSP_1:25</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
