:: Group and Field Definitions
:: by J\'ozef Bia{\l}as
::
:: Received October 27, 1989
:: Copyright (c) 1990-2012 Association of Mizar Users


begin

theorem Th1: :: REALSET1:1
for X, x being set
for F being Function of [:X,X:],X st x in [:X,X:] holds
F . x in X
proofend;

definition
let X be set ;
let F be BinOp of X;
mode Preserv of F ->  Subset of X means :Def1: :: REALSET1:def 1
 for x being set st x in [:it,it:] holds
F . x in it;
existence
 ex b1 being Subset of X st
for x being set st x in [:b1,b1:] holds
F . x in b1
proofend;
end;

:: deftheorem Def1 defines Preserv REALSET1:def_1_:_
 for X being set
for F being BinOp of X
for b3 being Subset of X holds
( b3 is Preserv of F iff for x being set st x in [:b3,b3:] holds
F . x in b3 );

definition
let R be Relation;
let A be set ;
funcR || A ->  set  equals :: REALSET1:def 2
R | [:A,A:];
coherence
 R | [:A,A:] is set ;
end;

:: deftheorem  defines || REALSET1:def_2_:_
 for R being Relation
for A being set holds R || A = R | [:A,A:];

registration
let R be Relation;
let A be set ;
clusterR || A ->  Relation-like ;
coherence
 R || A is Relation-like ;
end;

registration
let R be Function;
let A be set ;
clusterR || A ->  Function-like ;
coherence
 R || A is Function-like ;
end;

theorem Th2: :: REALSET1:2
for X being set
for F being BinOp of X
for A being Preserv of F holds F || A is BinOp of A
proofend;

definition
let X be set ;
let F be BinOp of X;
let A be Preserv of F;
:: original:||
redefine funcF || A ->  BinOp of A;
coherence
 F || A is BinOp of A by Th2;
end;

theorem Th3: :: REALSET1:3
for X being set holds
( not X is trivial iff for x being set holds X \ {x} is non empty set )
proofend;

theorem :: REALSET1:4
ex A being non empty set st
for z being Element of A holds A \ {z} is non empty set
proofend;

definition
let X be non trivial set ;
let F be BinOp of X;
let x be Element of X;
predF is_Bin_Op_Preserv x means :: REALSET1:def 3
( X \ {x} is Preserv of F & (F || X) \ {x} is BinOp of (X \ {x}) );
correctness
;
end;

:: deftheorem  defines is_Bin_Op_Preserv REALSET1:def_3_:_
 for X being non trivial set
for F being BinOp of X
for x being Element of X holds
( F is_Bin_Op_Preserv x iff ( X \ {x} is Preserv of F & (F || X) \ {x} is BinOp of (X \ {x}) ) );

theorem Th5: :: REALSET1:5
for X being set
for A being Subset of X ex F being BinOp of X st
for x being set st x in [:A,A:] holds
F . x in A
proofend;

definition
let X be set ;
let A be Subset of X;
mode Presv of X,A ->  BinOp of X means :Def4: :: REALSET1:def 4
 for x being set st x in [:A,A:] holds
it . x in A;
existence
 ex b1 being BinOp of X st
for x being set st x in [:A,A:] holds
b1 . x in A by Th5;
end;

:: deftheorem Def4 defines Presv REALSET1:def_4_:_
 for X being set
for A being Subset of X
for b3 being BinOp of X holds
( b3 is Presv of X,A iff for x being set st x in [:A,A:] holds
b3 . x in A );

theorem Th6: :: REALSET1:6
for X being set
for A being Subset of X
for F being Presv of X,A holds F || A is BinOp of A
proofend;

definition
let X be set ;
let A be Subset of X;
let F be Presv of X,A;
funcF ||| A ->  BinOp of A equals :: REALSET1:def 5
F || A;
coherence
 F || A is BinOp of A by Th6;
end;

:: deftheorem  defines ||| REALSET1:def_5_:_
 for X being set
for A being Subset of X
for F being Presv of X,A holds F ||| A = F || A;

definition
let A be set ;
let x be Element of A;
mode DnT of x,A ->  BinOp of A means :Def6: :: REALSET1:def 6
 for y being set st y in [:(A \ {x}),(A \ {x}):] holds
it . y in A \ {x};
existence
 ex b1 being BinOp of A st
for y being set st y in [:(A \ {x}),(A \ {x}):] holds
b1 . y in A \ {x} by Th5;
end;

:: deftheorem Def6 defines DnT REALSET1:def_6_:_
 for A being set
for x being Element of A
for b3 being BinOp of A holds
( b3 is DnT of x,A iff for y being set st y in [:(A \ {x}),(A \ {x}):] holds
b3 . y in A \ {x} );

theorem Th7: :: REALSET1:7
for A being non trivial set
for x being Element of A
for F being DnT of x,A holds F || (A \ {x}) is BinOp of (A \ {x})
proofend;

definition
let A be non trivial set ;
let x be Element of A;
let F be DnT of x,A;
funcF ! (A,x) ->  BinOp of (A \ {x}) equals :: REALSET1:def 7
F || (A \ {x});
coherence
 F || (A \ {x}) is BinOp of (A \ {x}) by Th7;
end;

:: deftheorem  defines ! REALSET1:def_7_:_
 for A being non trivial set
for x being Element of A
for F being DnT of x,A holds F ! (A,x) = F || (A \ {x});

theorem Th8: :: REALSET1:8
for F being non trivial set
for A being Singleton of F holds F \ A is non empty set
proofend;

registration
let F be non trivial set ;
let A be Singleton of F;
clusterF \ A ->  non empty ;
coherence
 not F \ A is empty by Th8;
end;