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

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 ) ) ;

( y in A & r > y ) or for y being ExtReal holds

( not y in A or not r > y ) ) ;

end;