let a, b be real number ; :: thesis: for X being Subset of R^1 st a < b & X = ].a,b.[ holds
Fr X = {a,b}
let X be Subset of R^1 ; :: thesis: ( a < b & X = ].a,b.[ implies Fr X = {a,b} )
assume that
A1: a < b and
A2: X = ].a,b.[ ; :: thesis: Fr X = {a,b}
A3: Cl X = Cl ].a,b.[ by A2, TOPREAL6:80
.= [.a,b.] by A1, TOPREAL6:82 ;
set RC = R^1 (right_closed_halfline b);
set LC = R^1 (left_closed_halfline a);
A4: R^1 (right_closed_halfline b) = right_closed_halfline b by TOPREALB:def 3;
A5: R^1 (left_closed_halfline a) = left_closed_halfline a by TOPREALB:def 3;
then A6: ].a,b.[ ` = (R^1 (left_closed_halfline a)) \/ (R^1 (right_closed_halfline b)) by A4, XXREAL_1:398;
A7: Cl (X ` ) = Cl (].a,b.[ ` ) by A2, TOPMETR:24, TOPREAL6:80
.= (Cl (left_closed_halfline a)) \/ (Cl (right_closed_halfline b)) by A4, A5, A6, Th3
.= (Cl (left_closed_halfline a)) \/ (right_closed_halfline b) by PSCOMP_1:34
.= (left_closed_halfline a) \/ (right_closed_halfline b) by PSCOMP_1:34 ;
[.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b} by A1, Th20;
hence Fr X = {a,b} by A3, A7, TOPS_1:def 2; :: thesis: verum