let a, b be Real; :: 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, JORDAN5A:24
.= [.a,b.] by A1, JORDAN5A:26 ;
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 & R^1 (left_closed_halfline a) = left_closed_halfline a ) by TOPREALB:def 3;
then A5: ].a,b.[ ` = (R^1 (left_closed_halfline a)) \/ (R^1 (right_closed_halfline b)) by XXREAL_1:398;
A6: [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b} by A1, Th8;
Cl (X `) = Cl (].a,b.[ `) by A2, JORDAN5A:24, TOPMETR:17
.= (Cl (left_closed_halfline a)) \/ (Cl (right_closed_halfline b)) by A4, A5, Th3
.= (Cl (left_closed_halfline a)) \/ (right_closed_halfline b) by MEASURE6:59
.= (left_closed_halfline a) \/ (right_closed_halfline b) by MEASURE6:59 ;
hence Fr X = {a,b} by A3, A6, TOPS_1:def 2; :: thesis: verum