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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font> holds  <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span></span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> implies <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:26_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">z1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:26_1"/><span class="lab"><font color="Green" title="E20">AA</font></span>: 
( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> &amp;  <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">z1</font> &amp; <font color="Maroon" title="c5">z1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ;<br/><a NAME="E4:26_1"/>
<font color="Maroon" title="c4">z</font> <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="c5">z1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</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="c5">z1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E20">AA</font></span></a>, <a class="txt" href="ndiff_5.html#E19"><span class="lab"><font color="Green" title="E18">LmX</font></span></a></span>;<br/><a NAME="E5:26_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E20">AA</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#D2" title="RLTOPSP1:def.2">RLTOPSP1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:26"/>
<span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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