let A be ext-real-membered set ; :: thesis: ( A is interval implies for r being ExtReal st inf A < r & r < sup A holds
r in A )
assume A1: A is interval ; :: thesis: for r being ExtReal st inf A < r & r < sup A holds
r in A
let r be ExtReal; :: thesis: ( inf A < r & r < sup A implies r in A )
assume that
A2: inf A < r and
A3: r < sup A ; :: thesis: r in A

per cases ( ex y being ExtReal st
( y in A & r > y ) or for y being ExtReal holds
( not y in A or not r > y ) ) ;

suppose ex y being ExtReal st
( y in A & r > y ) ; :: thesis: r in A

hence r in A by A1, A3, Th81; :: thesis: verum

end;

suppose for y being ExtReal holds
( not y in A or not r > y ) ; :: thesis: r in A

then r is LowerBound of A by Def2;
hence r in A by A2, Def4; :: thesis: verum

end;

end;