let r, s, a, b be real number ; :: thesis: ( r <= s implies for X being Subset of (Closed-Interval-TSpace r,s) st X = [.a,b.[ & r < a & b <= s holds
Int X = ].a,b.[ )
set L = Closed-Interval-TSpace r,s;
assume A1: r <= s ; :: thesis: for X being Subset of (Closed-Interval-TSpace r,s) st X = [.a,b.[ & r < a & b <= s holds
Int X = ].a,b.[
let X be Subset of (Closed-Interval-TSpace r,s); :: thesis: ( X = [.a,b.[ & r < a & b <= s implies Int X = ].a,b.[ )
assume that
A2: X = [.a,b.[ and
A3: r < a and
A4: b <= s ; :: thesis: Int X = ].a,b.[
set c = (r + a) / 2;
A5: r < (r + a) / 2 by A3, XREAL_1:228;
A6: (r + a) / 2 < a by A3, XREAL_1:228;
set C1 = R^1 ].((r + a) / 2),b.[;
reconsider A = X as Subset of R^1 by PRE_TOPC:39;
A7: R^1 ].((r + a) / 2),b.[ = ].((r + a) / 2),b.[ by TOPREALB:def 3;
A8: the carrier of (Closed-Interval-TSpace r,s) = [.r,s.] by A1, TOPMETR:25;
A9: R^1 ].((r + a) / 2),b.[ c= the carrier of (Closed-Interval-TSpace r,s)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in R^1 ].((r + a) / 2),b.[ or x in the carrier of (Closed-Interval-TSpace r,s) )
assume A10: x in R^1 ].((r + a) / 2),b.[ ; :: thesis: x in the carrier of (Closed-Interval-TSpace r,s)
then reconsider x = x as Real by A7;
( (r + a) / 2 < x & x < b ) by A7, A10, XXREAL_1:4;
then ( r <= x & x <= s ) by A4, A5, XXREAL_0:2;
hence x in the carrier of (Closed-Interval-TSpace r,s) by A8, XXREAL_1:1; :: thesis: verum

end;

A c= R^1 ].((r + a) / 2),b.[

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in A or x in R^1 ].((r + a) / 2),b.[ )
assume A11: x in A ; :: thesis: x in R^1 ].((r + a) / 2),b.[
then reconsider x = x as Real by A2;
( a <= x & x < b ) by A2, A11, XXREAL_1:3;
then ( (r + a) / 2 < x & x < b ) by A6, XXREAL_0:2;
hence x in R^1 ].((r + a) / 2),b.[ by A7, XXREAL_1:4; :: thesis: verum

end;

then Int A = Int X by A9, TOPS_3:57;
hence Int X = ].a,b.[ by A2, Th50; :: thesis: verum