let Gi be non trivial finite Subset of REAL ; :: thesis: for li, ri, ri' being Real st [li,ri] is Gap of Gi & [li,ri'] is Gap of Gi holds
ri = ri'
let li, ri, ri' be Real; :: thesis: ( [li,ri] is Gap of Gi & [li,ri'] is Gap of Gi implies ri = ri' )
A1: ( ri <= ri' & ri' <= ri implies ri = ri' ) by XXREAL_0:1;
assume A2: ( [li,ri] is Gap of Gi & [li,ri'] is Gap of Gi ) ; :: thesis: ri = ri'
then A3: ( ri in Gi & ri' in Gi ) by Th17;

per cases ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) by A2, Th17;

suppose A4: ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) ; :: thesis: ri = ri'

( ri' <= li or ( li < ri' & ri' < ri ) or ri <= ri' ) ;
hence ri = ri' by A1, A2, A3, A4, Th17; :: thesis: verum

end;

suppose A5: ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ; :: thesis: ri = ri'

( ri' < ri or ( ri <= ri' & ri' <= li ) or li < ri' ) ;
hence ri = ri' by A1, A2, A3, A5, Th17; :: thesis: verum

end;

end;