:: REALSET1 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

theorem Th1: :: REALSET1:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 :
for X being set
for F being BinOp of X
for A being Subset of X holds
( F is_in A iff for x being set st x in [:A,A:] holds
F . x in A );

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
ex b1 being Subset of X st
for x being set st x in [:b1,b1:] holds
F . x in b1
by Th2;
end;

:: deftheorem Def2 defines Preserv REALSET1:def 2 :
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 ;
func R || A -> set equals :: REALSET1:def 3
R | [:A,A:];
coherence
R | [:A,A:] is set
;
end;

:: deftheorem defines || REALSET1:def 3 :
for R being Relation
for A being set holds R || A = R | [:A,A:];

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

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

theorem Th3: :: REALSET1:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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
F || A is BinOp of A
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 :
for IT being set holds
( IT is trivial iff ( IT is empty or ex x being set st IT = {x} ) );

registration
cluster non empty trivial set ;
existence
ex b1 being set st
( b1 is trivial & not b1 is empty )
proof end;
cluster non empty non trivial set ;
existence
ex b1 being set st
( not b1 is trivial & not b1 is empty )
proof end;
cluster non trivial -> non empty set ;
coherence
for b1 being set st not b1 is trivial holds
not b1 is empty
proof end;
end;

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

theorem Th4: :: REALSET1:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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 :
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 Th8: :: REALSET1:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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
ex b1 being BinOp of X st
for x being set st x in [:A,A:] holds
b1 . x in A
by Th8;
end;

:: deftheorem Def6 defines Presv REALSET1:def 6 :
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 Th9: :: REALSET1:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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
F || A is BinOp of A
by Th9;
end;

:: deftheorem defines ||| REALSET1:def 7 :
for X being set
for A being Subset of X
for F being Presv of X,A holds F ||| A = F || A;

theorem Th10: :: REALSET1:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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
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 Th10;
end;

:: deftheorem Def8 defines DnT REALSET1:def 8 :
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 Th11: :: REALSET1:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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
F || (A \ {x}) is BinOp of A \ {x}
by Th11;
end;

:: deftheorem defines ! REALSET1:def 9 :
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});

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
ex b1 being set ex x being Element of F st b1 = {x}
proof end;
end;

:: deftheorem Def10 defines OnePoint REALSET1:def 10 :
for F being non trivial set
for b2 being set holds
( b2 is OnePoint of F iff ex x being Element of F st b2 = {x} );

theorem Th12: :: REALSET1:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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
not F \ A is empty
by Th12;
end;

registration
let F be non trivial set ;
cluster non empty OnePoint of F;
existence
not for b1 being OnePoint of F holds b1 is empty
proof end;
end;

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

theorem :: REALSET1:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for A being finite set holds
( A is trivial iff card A < 2 )
proof end;

theorem Th14: :: REALSET1:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
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;