let x, y be ExtReal; :: thesis: ( x < y implies inf ].x,y.] = x )
assume A1: x < y ; :: thesis: inf ].x,y.] = x
A2: for z being LowerBound of ].x,y.] holds z <= x

proof

let z be LowerBound of ].x,y.]; :: thesis: z <= x
for r being ExtReal st x < r & r < y holds
z <= r

proof

let r be ExtReal; :: thesis: ( x < r & r < y implies z <= r )
assume that
A3: x < r and
A4: r < y ; :: thesis: z <= r
r in ].x,y.] by A3, A4, XXREAL_1:2;
hence z <= r by Def2; :: thesis: verum

end;

hence z <= x by A1, XREAL_1:228; :: thesis: verum

end;

x is LowerBound of ].x,y.] by Th18;
hence inf ].x,y.] = x by A2, Def4; :: thesis: verum