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;

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

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 )

now :: thesis: 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;

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 )

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;

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 )

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;

end;

end;

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

end;

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 )

now :: thesis: 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;

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 )

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;

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 )

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;

end;

end;

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

end;

end;