let f be non constant standard special_circular_sequence; :: thesis: for a, b, c being set st a in (L~ f) ` & b in (L~ f) ` & c in (L~ f) ` & ( for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not a in C or not b in C ) ) & ( for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not b in C or not c in C ) ) holds

ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

let a, b, c be set ; :: thesis: ( a in (L~ f) ` & b in (L~ f) ` & c in (L~ f) ` & ( for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not a in C or not b in C ) ) & ( for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not b in C or not c in C ) ) implies ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C ) )

assume that

A1: a in (L~ f) ` and

A2: b in (L~ f) ` and

A3: c in (L~ f) ` and

A4: for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not a in C or not b in C ) and

A5: for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not b in C or not c in C ) ; :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

A6: LeftComp f is_a_component_of (L~ f) ` by GOBOARD9:def 1;

A7: RightComp f is_a_component_of (L~ f) ` by GOBOARD9:def 2;

( not C is_a_component_of (L~ f) ` or not a in C or not b in C ) ) & ( for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not b in C or not c in C ) ) holds

ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

let a, b, c be set ; :: thesis: ( a in (L~ f) ` & b in (L~ f) ` & c in (L~ f) ` & ( for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not a in C or not b in C ) ) & ( for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not b in C or not c in C ) ) implies ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C ) )

assume that

A1: a in (L~ f) ` and

A2: b in (L~ f) ` and

A3: c in (L~ f) ` and

A4: for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not a in C or not b in C ) and

A5: for C being Subset of (TOP-REAL 2) holds

( not C is_a_component_of (L~ f) ` or not b in C or not c in C ) ; :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

A6: LeftComp f is_a_component_of (L~ f) ` by GOBOARD9:def 1;

A7: RightComp f is_a_component_of (L~ f) ` by GOBOARD9:def 2;

per cases
( ( a in LeftComp f & b in RightComp f ) or ( a in RightComp f & b in LeftComp f ) )
by A1, A2, A4, Th15;

end;

suppose A8:
( a in LeftComp f & b in RightComp f )
; :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

( C is_a_component_of (L~ f) ` & a in C & c in C ) ; :: thesis: verum

end;

( C is_a_component_of (L~ f) ` & a in C & c in C )

now :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )end;

hence
ex C being Subset of (TOP-REAL 2) st ( C is_a_component_of (L~ f) ` & a in C & c in C )

per cases
( ( b in LeftComp f & c in RightComp f ) or ( b in RightComp f & c in LeftComp f ) )
by A2, A3, A5, Th15;

end;

suppose
( b in LeftComp f & c in RightComp f )
; :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

( C is_a_component_of (L~ f) ` & a in C & c in C )

then
LeftComp f meets RightComp f
by A8, XBOOLE_0:3;

hence ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C ) by GOBRD14:14; :: thesis: verum

end;hence ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C ) by GOBRD14:14; :: thesis: verum

suppose
( b in RightComp f & c in LeftComp f )
; :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

( C is_a_component_of (L~ f) ` & a in C & c in C )

hence
ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C ) by A6, A8; :: thesis: verum

end;( C is_a_component_of (L~ f) ` & a in C & c in C ) by A6, A8; :: thesis: verum

( C is_a_component_of (L~ f) ` & a in C & c in C ) ; :: thesis: verum

suppose A9:
( a in RightComp f & b in LeftComp f )
; :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

( C is_a_component_of (L~ f) ` & a in C & c in C ) ; :: thesis: verum

end;

( C is_a_component_of (L~ f) ` & a in C & c in C )

now :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )end;

hence
ex C being Subset of (TOP-REAL 2) st ( C is_a_component_of (L~ f) ` & a in C & c in C )

per cases
( ( b in RightComp f & c in LeftComp f ) or ( b in LeftComp f & c in RightComp f ) )
by A2, A3, A5, Th15;

end;

suppose
( b in RightComp f & c in LeftComp f )
; :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

( C is_a_component_of (L~ f) ` & a in C & c in C )

then
LeftComp f meets RightComp f
by A9, XBOOLE_0:3;

hence ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C ) by GOBRD14:14; :: thesis: verum

end;hence ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C ) by GOBRD14:14; :: thesis: verum

suppose
( b in LeftComp f & c in RightComp f )
; :: thesis: ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C )

( C is_a_component_of (L~ f) ` & a in C & c in C )

hence
ex C being Subset of (TOP-REAL 2) st

( C is_a_component_of (L~ f) ` & a in C & c in C ) by A7, A9; :: thesis: verum

end;( C is_a_component_of (L~ f) ` & a in C & c in C ) by A7, A9; :: thesis: verum

( C is_a_component_of (L~ f) ` & a in C & c in C ) ; :: thesis: verum