let C be non empty connected compact Subset of R^1 ; :: thesis: C is non empty closed-interval Subset of REAL
reconsider C' = C as non empty Subset of REAL by TOPMETR:24;
C is closed by COMPTS_1:16;
then A1: C' is closed by TOPREAL6:79;
A2: ( C' is bounded_below & C' is bounded_above ) by Th107;
then A3: inf C' in C' by A1, RCOMP_1:31;
sup C' in C' by A1, A2, RCOMP_1:30;
then A4: [.(inf C'),(sup C').] = C' by A3, Th109, Th111;
C' is bounded by A2, XXREAL_2:def 11;
then lower_bound C' <= upper_bound C' by SEQ_4:24;
hence C is non empty closed-interval Subset of REAL by A4, INTEGRA1:def 1; :: thesis: verum