let C be non empty connected compact Subset of R^1 ; :: thesis: C is non empty closed-interval Subset of REAL
reconsider C9 = C as non empty Subset of REAL by TOPMETR:24;
C is closed by COMPTS_1:16;
then A1: C9 is closed by TOPREAL6:79;
then A2: upper_bound C9 in C9 by Th107, RCOMP_1:30;
( C9 is bounded_below & C9 is bounded_above ) by Th107;
then C9 is bounded by XXREAL_2:def 11;
then A3: lower_bound C9 <= upper_bound C9 by SEQ_4:24;
lower_bound C9 in C9 by A1, Th107, RCOMP_1:31;
then [.(lower_bound C9),(upper_bound C9).] = C9 by A2, Th109, Th111;
hence C is non empty closed-interval Subset of REAL by A3, INTEGRA1:def 1; :: thesis: verum