let Gi be non trivial finite Subset of REAL; :: thesis: for li, ri, li9 being Real st [li,ri] is Gap of Gi & [li9,ri] is Gap of Gi holds
li = li9
let li, ri, li9 be Real; :: thesis: ( [li,ri] is Gap of Gi & [li9,ri] is Gap of Gi implies li = li9 )
A1: ( li <= li9 & li9 <= li implies li = li9 ) by XXREAL_0:1;
assume that
A2: [li,ri] is Gap of Gi and
A3: [li9,ri] is Gap of Gi ; :: thesis: li = li9
A4: li in Gi by A2, Th13;
A5: li9 in Gi by A3, Th13;
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, Th13;
suppose A6: ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) ; :: thesis: li = li9
( li9 <= li or ( li < li9 & li9 < ri ) or ri <= li9 ) ;
hence li = li9 by A1, A3, A4, A5, A6, Th13; :: thesis: verum
end;
suppose A7: ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ; :: thesis: li = li9
( li9 < ri or ( ri <= li9 & li9 <= li ) or li < li9 ) ;
hence li = li9 by A1, A3, A4, A5, A7, Th13; :: thesis: verum
end;
end;