let s, t, r be ext-real number ; :: thesis: ( s < t implies ].r,t.[ \ [.s,t.[ = ].r,s.[ )
assume A1: s < t ; :: thesis: ].r,t.[ \ [.s,t.[ = ].r,s.[
let p be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not p in ].r,t.[ \ [.s,t.[ or p in ].r,s.[ ) & ( not p in ].r,s.[ or p in ].r,t.[ \ [.s,t.[ ) )
thus ( p in ].r,t.[ \ [.s,t.[ implies p in ].r,s.[ ) :: thesis: ( not p in ].r,s.[ or p in ].r,t.[ \ [.s,t.[ )

proof

assume A2: p in ].r,t.[ \ [.s,t.[ ; :: thesis: p in ].r,s.[
then A3: not p in [.s,t.[ by XBOOLE_0:def 5;
A4: r < p by A2, Th4;
( p < s or t <= p ) by A3, Th3;
hence p in ].r,s.[ by A2, A4, Th4; :: thesis: verum

end;

assume A5: p in ].r,s.[ ; :: thesis: p in ].r,t.[ \ [.s,t.[
then A6: p < s by Th4;
A7: r < p by A5, Th4;
p < t by A1, A6, XXREAL_0:2;
then A8: p in ].r,t.[ by A7, Th4;
not p in [.s,t.[ by A6, Th3;
hence p in ].r,t.[ \ [.s,t.[ by A8, XBOOLE_0:def 5; :: thesis: verum