let A be ext-real-membered set ; :: thesis: ( A is interval implies for x, r being ExtReal st x in A & inf A < r & r <= x holds

r in A )

assume A1: A is interval ; :: thesis: for x, r being ExtReal st x in A & inf A < r & r <= x holds

r in A

let x, r be ExtReal; :: thesis: ( x in A & inf A < r & r <= x implies r in A )

assume that

A2: x in A and

A3: inf A < r and

A4: r <= x ; :: thesis: r in A

r in A )

assume A1: A is interval ; :: thesis: for x, r being ExtReal st x in A & inf A < r & r <= x holds

r in A

let x, r be ExtReal; :: thesis: ( x in A & inf A < r & r <= x implies r in A )

assume that

A2: x in A and

A3: inf A < r and

A4: r <= x ; :: 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;