:: REALSET1 semantic presentation

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

proof end;

theorem Th2: :: REALSET1:2

for X being set
for F being BinOp of X ex A being Subset of X st
for x being set st x in [:A,A:] holds
F . x in A

proof end;

definition

let X be set ;
let F be BinOp of X;
let A be Subset of X;
pred F is_in A means :: REALSET1:def 1
for x being set st x in [:A,A:] holds
F . x in A;

end;

:: deftheorem defines is_in REALSET1:def 1 :

definition

let X be set ;
let F be BinOp of X;
mode Preserv of F -> Subset of X means :Def2: :: REALSET1:def 2
for x being set st x in [:it,it:] holds
F . x in it;
existence by Th2;

end;

:: deftheorem Def2 defines Preserv REALSET1:def 2 :

definition

let R be Relation;
let A be set ;
func R || A -> set equals :: REALSET1:def 3
R | [:A,A:];
coherence ;

end;

:: deftheorem defines || REALSET1:def 3 :

registration

let R be Relation;
let A be set ;
cluster R || A -> Relation-like ;
coherence ;

end;

registration

let R be Function;
let A be set ;
cluster R || A -> Relation-like Function-like ;
coherence ;

end;

theorem Th3: :: REALSET1:3

for X being set
for F being BinOp of X
for A being Preserv of F holds F || A is BinOp of A

proof end;

definition

let X be set ;
let F be BinOp of X;
let A be Preserv of F;
:: original: ||
redefine func F || A -> BinOp of A;
coherence by Th3;

end;

definition

let IT be set ;
attr IT is trivial means :Def4: :: REALSET1:def 4
( IT is empty or ex x being set st IT = {x} );

end;

:: deftheorem Def4 defines trivial REALSET1:def 4 :

registration

cluster non empty trivial set ;
existence

proof end;

cluster non empty non trivial set ;
existence

proof end;

cluster non trivial -> non empty set ;
coherence

proof end;

end;

registration

let y be set ;
cluster {y} -> trivial ;
coherence by Def4;

end;

theorem Th4: :: REALSET1:4

for X being non empty set holds
( not X is trivial iff for x being set holds X \ {x} is non empty set )

proof end;

theorem :: REALSET1:5

ex A being non empty set st
for z being Element of A holds A \ {z} is non empty set

proof end;

theorem Th6: :: REALSET1:6

for X being non empty set st ( for x being Element of X holds X \ {x} is non empty set ) holds
not X is trivial

proof end;

theorem :: REALSET1:7

for A being non empty set st ( for x being Element of A holds A \ {x} is non empty set ) holds
A is non trivial set by Th6;

definition

let X be non trivial set ;
let F be BinOp of X;
let x be Element of X;
pred F is_Bin_Op_Preserv x means :: REALSET1:def 5
( X \ {x} is Preserv of F & (F || X) \ {x} is BinOp of X \ {x} );
correctness ;

end;

:: deftheorem defines is_Bin_Op_Preserv REALSET1:def 5 :

theorem Th8: :: REALSET1:8

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

proof end;

definition

let X be set ;
let A be Subset of X;
mode Presv of X,A -> BinOp of X means :Def6: :: REALSET1:def 6
for x being set st x in [:A,A:] holds
it . x in A;
existence by Th8;

end;

:: deftheorem Def6 defines Presv REALSET1:def 6 :

theorem Th9: :: REALSET1:9

for X being set
for A being Subset of X
for F being Presv of X,A holds F || A is BinOp of A

proof end;

definition

let X be set ;
let A be Subset of X;
let F be Presv of X,A;
func F ||| A -> BinOp of A equals :: REALSET1:def 7
F || A;
coherence by Th9;

end;

:: deftheorem defines ||| REALSET1:def 7 :

theorem Th10: :: REALSET1:10

for A being set
for x being Element of A ex F being BinOp of A st
for y being set st y in [:(A \ {x}),(A \ {x}):] holds
F . y in A \ {x}

proof end;

definition

let A be set ;
let x be Element of A;
mode DnT of x,A -> BinOp of A means :Def8: :: REALSET1:def 8
for y being set st y in [:(A \ {x}),(A \ {x}):] holds
it . y in A \ {x};
existence by Th10;

end;

:: deftheorem Def8 defines DnT REALSET1:def 8 :

theorem Th11: :: REALSET1:11

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}

proof end;

definition

let A be non trivial set ;
let x be Element of A;
let F be DnT of x,A;
func F ! A,x -> BinOp of A \ {x} equals :: REALSET1:def 9
F || (A \ {x});
coherence by Th11;

end;

:: deftheorem defines ! REALSET1:def 9 :

definition

let F be non trivial set ;
mode OnePoint of F -> set means :Def10: :: REALSET1:def 10
ex x being Element of F st it = {x};
existence

proof end;

end;

:: deftheorem Def10 defines OnePoint REALSET1:def 10 :

theorem Th12: :: REALSET1:12

for F being non trivial set
for A being OnePoint of F holds F \ A is non empty set

proof end;

registration

let F be non trivial set ;
let A be OnePoint of F;
cluster F \ A -> non empty ;
coherence by Th12;

end;

registration

let F be non trivial set ;
cluster non empty OnePoint of F;
existence

proof end;

end;

definition

let F be non trivial set ;
let x be Element of F;
:: original: {
redefine func {x} -> OnePoint of F;
coherence by Def10;

end;

theorem :: REALSET1:13

for A being finite set holds
( A is trivial iff card A < 2 )

proof end;

theorem Th14: :: REALSET1:14

for A being set holds
( not A is trivial iff ex a1, a2 being set st
( a1 in A & a2 in A & a1 <> a2 ) )

proof end;

theorem :: REALSET1:15

for D being set
for A being Subset of D holds
( not A is trivial iff ex d1, d2 being Element of D st
( d1 in A & d2 in A & d1 <> d2 ) )

proof end;