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<span class="kw">let </span><font color="Maroon" title="c3">I1</font>, <font color="Maroon" title="c4">I2</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp;  not <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">I1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp;  not <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">I2</font> implies <font color="Maroon" title="c3">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_1_2"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:19_1_2"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
 not <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">I1</font>
 <span class="kw">and </span><br/><a NAME="E3:19_1_2"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:19_1_2"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
 not <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">I2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I2</font></span><br/>

<a NAME="E5:19_1_2"/>
( <font color="Maroon" title="c3">I1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c4">I2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E3:19_1_2"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">i11</font> = <font color="Maroon" title="c3">I1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1, <font color="Maroon" title="c6">i21</font> = <font color="Maroon" title="c4">I2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E7:19_1_2"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<font color="Maroon" title="c3">I1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">I2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_1_2_1"/>
<font color="Maroon" title="c3">I1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">I2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A11</font></span>: 
<font color="Maroon" title="c3">I1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i21</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">i1</font> = <font color="Maroon" title="c3">I1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<a NAME="E4:19_1_2_1"/>
<span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c6">i21</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c7">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_2_1"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="prob_1.html#D5" onmouseover="rs('prob_1/D5')" onmouseout="rh()">PROB_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:19_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_2"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E3:19_1_2"><span class="lab"><font color="Green" title="E12">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E8:19_1_2"/>
<font color="Maroon" title="c4">I2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">I1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_1_2_2"/>
<font color="Maroon" title="c3">I1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">I2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<font color="Maroon" title="c4">I2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">i11</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">i2</font> = <font color="Maroon" title="c4">I2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<a NAME="E4:19_1_2_2"/>
<span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c5">i11</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c7">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_2_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="prob_1.html#D5" onmouseover="rs('prob_1/D5')" onmouseout="rh()">PROB_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:19_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E4:19_1_2"><span class="lab"><font color="Green" title="E13">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E9:19_1_2"/>
<font color="Maroon" title="c3">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
