let Gi be non trivial finite Subset of REAL ; :: thesis: for li, ri, li' being Real st [li,ri] is Gap of Gi & [li',ri] is Gap of Gi holds
li = li'
let li, ri, li' be Real; :: thesis: ( [li,ri] is Gap of Gi & [li',ri] is Gap of Gi implies li = li' )
A1: ( li <= li' & li' <= li implies li = li' ) by XXREAL_0:1;
assume that
A2: [li,ri] is Gap of Gi and
A3: [li',ri] is Gap of Gi ; :: thesis: li = li'
A4: li in Gi by A2, Th17;
A5: li' in Gi by A3, 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 A6: ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) ; :: thesis: li = li'

( li' <= li or ( li < li' & li' < ri ) or ri <= li' ) ;
hence li = li' by A1, A3, A4, A5, A6, Th17; :: thesis: verum

end;

suppose A7: ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ; :: thesis: li = li'

( li' < ri or ( ri <= li' & li' <= li ) or li < li' ) ;
hence li = li' by A1, A3, A4, A5, A7, Th17; :: thesis: verum

end;

end;