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<span class="kw">let </span><font color="Maroon" title="c1">E</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> <br/> for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">LO</font> being  <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">E</font> holds  <font color="Olive" title="b3">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b2">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">LO</font> being  <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">E</font> holds  <font color="Olive" title="b2">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">LO</font> being  <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">E</font> holds  <font color="Olive" title="b1">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c3">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">LO</font> be   <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">E</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="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c3">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>







<span class="kw">consider </span><font color="Maroon" title="c5">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c5">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Olive" title="b2">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">s9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E4:7"/>
<font color="Maroon" title="c3">s</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">s9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">and </span><br/><a NAME="E5:7"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c6">s9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>;<br/>
<a NAME="E6:7"/>
<font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c6">s9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<a NAME="E7:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c6">s9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E8:7"/>
<font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<a NAME="E9:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">E</font> = 
<font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="group_1.html#T10" title="GROUP_1:th.10">GROUP_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c6">s9</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c3">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D1" target="_self" title="GROUP_10:def.1">Def1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:7"/>
<font color="Maroon" title="c4">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c3">s</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="#E9:7"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</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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