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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">  for <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">G1</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b3">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">G</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp; <font color="Olive" title="b2">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Olive" title="b1">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Olive" title="b2">G1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> holds <br/>( <font color="Olive" title="b1">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">G1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">G1</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</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">  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c2">G</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp; <font color="Maroon" title="c3">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/>( <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</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">G</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp; <font color="Maroon" title="c3">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> implies ( <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c2">G</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a> 
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c3">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a> 
 <span class="kw">and </span><br/><a NAME="E3:31"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E4:31"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</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="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<a NAME="E5:31"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font> holds <br/>( <font color="Olive" title="b1">A</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</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="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font> implies ( <font color="Maroon" title="c5">A</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</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="c5">A</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<a NAME="E2:31_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G</font> or <font color="Maroon" title="c5">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_1"/>
( <font color="Maroon" title="c5">A</font> is  <a href="topdim_1.html#NV2" title="TOPDIM_1:NV.2">finite-ind</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="topdim_1.html#T11" target="_self" onmouseover="rs('topdim_1/T11')" onmouseout="rh()">Th11</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="E6:31"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="topdim_1.html#T11" target="_self" onmouseover="rs('topdim_1/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:31"/>
( <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font> is  <a href="topdim_1.html#NV4" title="TOPDIM_1:NV.4">finite-ind</a>  &amp;  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="topdim_1.html#T11" target="_self" onmouseover="rs('topdim_1/T11')" onmouseout="rh()">Th11</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/>


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