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<span class="kw">let </span><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</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"> (  ex <font color="Olive" title="b1">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Olive" title="b1">F</font></span>)</span> ) &amp;  ex <font color="Olive" title="b1">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Olive" title="b1">F</font></span>)</span> ) implies <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><a NAME="E2:17_1_2"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c6">F1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E3:17_1_2"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_2"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">F1</font></span>)</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"> (  for <font color="Olive" title="b1">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>  holds <br/>(  not <font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  or  not  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font> or  not <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Olive" title="b1">F</font></span>)</span> ) or <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c7">F2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><a NAME="E6:17_1_2"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c7">F2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E7:17_1_2"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">and </span><br/><a NAME="E8:17_1_2"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c5">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</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"> <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span><span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E9:17_1_2"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E6:17_1_2"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E7:17_1_2"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="finseq_1.html#T48" onmouseover="rs('finseq_1/T48')" onmouseout="rh()">FINSEQ_1:48</a></span>;<br/>
<a NAME="E10:17_1_2"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:17_1_2"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:17_1_2"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</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">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<a NAME="E2:17_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E7:17_1_2"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E11:17_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:17_1_2_1"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="funct_1.html#T74" onmouseover="rs('funct_1/T74')" onmouseout="rh()">FUNCT_1:74</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">y</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="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>] )</span><br/>

<a NAME="E5:17_1_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_2_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:17_1_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E7:17_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_2_1"><span class="lab"><font color="Green" title="E23">A14</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">consider </span><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:17_1_2"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E12:17_1_2"><span class="lab"><font color="Green" title="E22">A13</font></span></a>);<br/></span>
<a NAME="E15:17_1_2"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:17_1_2_2"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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">  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:17_1_2_2"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</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="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font></span><br/>

<a NAME="E3:17_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E7:17_1_2"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:17_1_2_2"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">and </span><br/><a NAME="E6:17_1_2_2"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:17_1_2_2"/>
<font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K9" title="RELAT_1:func.9">Im</a> (<font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c9">y</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2_2"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E6:17_1_2_2"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="funct_1.html#T59" onmouseover="rs('funct_1/T59')" onmouseout="rh()">FUNCT_1:59</a></span>;<br/>
<a NAME="E8:17_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="funct_1.html#T82" onmouseover="rs('funct_1/T82')" onmouseout="rh()">FUNCT_1:82</a></span>;<br/>
<a NAME="E9:17_1_2_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E11:17_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E14:17_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E5:17_1_2_2"><span class="lab"><font color="Green" title="E26">A18</font></span></a></span>;<br/>
<a NAME="E10:17_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T18" onmouseover="rs('zfmisc_1/T18')" onmouseout="rh()">ZFMISC_1:18</a></span>;<br/>
<a NAME="E11:17_1_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E11:17_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E5:17_1_2_2"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:17_1_2_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:17_1_2_2"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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">reconsider </span><font color="Maroon" title="c9">G1</font> = <font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">F1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<a NAME="E17:17_1_2"/><span class="lab"><font color="Green" title="E25">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D5" target="_self" onmouseover="rs('vectsp_6/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E18:17_1_2"/><span class="lab"><font color="Green" title="E26">A22</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c11">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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"> (  not <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">f</font> or  not <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">f</font> or  not <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y2</font> or <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17_1_2_3"/><span class="lab"><font color="Green" title="E27">A23</font></span>: 
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E2:17_1_2_3"/><span class="lab"><font color="Green" title="E28">A24</font></span>: 
<font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E3:17_1_2_3"/><span class="lab"><font color="Green" title="E29">A25</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y2</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="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font></span><br/>

<a NAME="E4:17_1_2_3"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" onmouseover="rs('funct_2/T52')" onmouseout="rh()">FUNCT_2:52</a></span>;<br/>
<a NAME="E5:17_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A26</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_2_3"><span class="lab"><font color="Green" title="E27">A23</font></span></a></span>;<br/>
<a NAME="E6:17_1_2_3"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
<font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_2_3"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E2:17_1_2_3"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:17_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A28</font></span>: 
<font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:17_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a></span>;<br/>
<a NAME="E8:17_1_2_3"/><span class="lab"><font color="Green" title="E33">A29</font></span>: 
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_2_3"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E1:17_1_2_3"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:17_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E7:17_1_2"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E11:17_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E10:17_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A30</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<a NAME="E11:17_1_2_3"/>
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E7:17_1_2"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E11:17_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E6:17_1_2_3"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E12:17_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A31</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<a NAME="E13:17_1_2_3"/>
<font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:17_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E8:17_1_2_3"><span class="lab"><font color="Green" title="E33">A29</font></span></a></span>;<br/>
<a NAME="E14:17_1_2_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2_3"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E7:17_1_2_3"><span class="lab"><font color="Green" title="E32">A28</font></span></a>, <a class="txt" href="#E10:17_1_2_3"><span class="lab"><font color="Green" title="E34">A30</font></span></a>, <a class="txt" href="#E12:17_1_2_3"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="ref" href="funct_1.html#T91" onmouseover="rs('funct_1/T91')" onmouseout="rh()">FUNCT_1:91</a></span>;<br/>
<a NAME="E15:17_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A32</font></span>: 
<font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T3" onmouseover="rs('zfmisc_1/T3')" onmouseout="rh()">ZFMISC_1:3</a></span>;<br/>
<a NAME="E16:17_1_2_3"/>
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">F2</font> &amp; <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">F2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E11:17_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E5:17_1_2_3"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E1:17_1_2_3"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="txt" href="#E2:17_1_2_3"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:17_1_2_3"/>
<font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_2"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E15:17_1_2_3"><span class="lab"><font color="Green" title="E36">A32</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">set </span><font color="Maroon" title="c10">G2</font> = <font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font>;<br/>
<a NAME="E19:17_1_2"/><span class="lab"><font color="Green" title="E27">A33</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:17_1_2"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E18:17_1_2"><span class="lab"><font color="Green" title="E26">A22</font></span></a>, <a class="ref" href="funct_2.html#T57" onmouseover="rs('funct_2/T57')" onmouseout="rh()">FUNCT_2:57</a></span>;<br/>
<a NAME="E21:17_1_2"/>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F1</font></span>)</span> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">f</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">G1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1_2"><span class="lab"><font color="Green" title="E25">A21</font></span></a></span>;<br/>
<a NAME="E23:17_1_2"/><span class="lab"><font color="Green" title="E28">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D5" target="_self" onmouseover="rs('vectsp_6/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E24:17_1_2"/><span class="lab"><font color="Green" title="E29">A35</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_2_4"><span class="kw">now </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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c13">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:17_1_2_4"/><span class="lab"><font color="Green" title="E30">A36</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><br/><a NAME="E2:17_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A37</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font></span>)</span> &amp; <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:17_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E11:17_1_2"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E19:17_1_2"><span class="lab"><font color="Green" title="E27">A33</font></span></a>, <a class="ref" href="vectsp_6.html#D5" target="_self" onmouseover="rs('vectsp_6/D5')" onmouseout="rh()">Def5</a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:17_1_2_4"/>
<font color="Maroon" title="c13">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:17_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E1:17_1_2_4"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">u</font> = <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T11" onmouseover="rs('finseq_2/T11')" onmouseout="rh()">FINSEQ_2:11</a></span>;<br/><a NAME="E5:17_1_2_4"/>
<font color="Maroon" title="c13">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E17:17_1_2"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E23:17_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E24:17_1_2"><span class="lab"><font color="Green" title="E29">A35</font></span></a>, <a class="txt" href="#E19:17_1_2"><span class="lab"><font color="Green" title="E27">A33</font></span></a>, <a class="txt" href="#E1:17_1_2_4"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:17_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A38</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:17_1_2"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">m</font> = <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</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="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_2_4"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="finseq_2.html#T11" onmouseover="rs('finseq_2/T11')" onmouseout="rh()">FINSEQ_2:11</a></span>;<br/><span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
 <a href="relat_1.html#K9" title="RELAT_1:func.9">Im</a> (<font color="Maroon" title="c6">F1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_2_4"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="funct_1.html#T59" onmouseover="rs('funct_1/T59')" onmouseout="rh()">FUNCT_1:59</a></span>
<br/>.= 
<font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E23:17_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E19:17_1_2"><span class="lab"><font color="Green" title="E27">A33</font></span></a>, <a class="txt" href="#E14:17_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E1:17_1_2_4"><span class="lab"><font color="Green" title="E30">A36</font></span></a></span>
;<br/><a NAME="E10:17_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A39</font></span>: 
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T75" onmouseover="rs('funct_1/T75')" onmouseout="rh()">FUNCT_1:75</a>, <a class="ref" href="zfmisc_1.html#T18" onmouseover="rs('zfmisc_1/T18')" onmouseout="rh()">ZFMISC_1:18</a></span>;<br/><a NAME="E11:17_1_2_4"/>
<font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_2_4"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E12:17_1_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1_2"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E10:17_1_2"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E24:17_1_2"><span class="lab"><font color="Green" title="E29">A35</font></span></a>, <a class="txt" href="#E6:17_1_2_4"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="txt" href="#E10:17_1_2_4"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="txt" href="#E2:17_1_2_4"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="ref" href="vectsp_6.html#D5" target="_self" onmouseover="rs('vectsp_6/D5')" onmouseout="rh()">Def5</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="E26:17_1_2"/>
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E4:17_1_2"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E6:17_1_2"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E7:17_1_2"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E8:17_1_2"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E17:17_1_2"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E23:17_1_2"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="finseq_1.html#T48" onmouseover="rs('finseq_1/T48')" onmouseout="rh()">FINSEQ_1:48</a>, <a class="ref" href="rlvect_2.html#T6" onmouseover="rs('rlvect_2/T6')" onmouseout="rh()">RLVECT_2:6</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>
