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<span class="kw">let </span><font color="Maroon" title="c1">g</font>, <font color="Maroon" title="c2">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">g</font> implies ( <font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:60"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</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="E2:60"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<a NAME="E3:60"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:60"/>
<font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="funct_1.html#T47" target="_self" title="FUNCT_1:th.47">Th47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<span class="kw">assume </span><a NAME="E5:60"/>
not <font color="Maroon" title="c1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:60"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">g</font>
 <span class="kw">and </span><br/><a NAME="E8:60"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c4">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">g</font>
 <span class="kw">and </span><br/><a NAME="E9:60"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
( <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y2</font> &amp; <font color="Maroon" title="c3">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" target="_self" title="FUNCT_1:def.8">Def8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:60"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E12:60"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E8:60"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="funct_1.html#D5" target="_self" title="FUNCT_1:def.5">Def5</a></span>;<br/>
<a NAME="E13:60"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="funct_1.html#T23" target="_self" title="FUNCT_1:th.23">Th23</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:60"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E16:60"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E7:60"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="funct_1.html#D5" target="_self" title="FUNCT_1:def.5">Def5</a></span>;<br/>
<a NAME="E17:60"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:60"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="funct_1.html#T23" target="_self" title="FUNCT_1:th.23">Th23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:60"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E9:60"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E15:60"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E16:60"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E11:60"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E12:60"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E13:60"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="funct_1.html#D8" target="_self" title="FUNCT_1:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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