let A be ext-real-membered set ; :: thesis: ( A is connected implies for r being ext-real number st inf A < r & r < sup A holds
r in A )
assume Z: A is connected ; :: thesis: for r being ext-real number st inf A < r & r < sup A holds
r in A
let r be ext-real number ; :: thesis: ( inf A < r & r < sup A implies r in A )
assume that
Z4: inf A < r and
Z3: r < sup A ; :: thesis: r in A

per cases ( ex y being ext-real number st
( y in A & r > y ) or for y being ext-real number holds
( not y in A or not r > y ) ) ;

suppose ex y being ext-real number st
( y in A & r > y ) ; :: thesis: r in A

hence r in A by Z, Z3, Th85; :: thesis: verum

end;

suppose for y being ext-real number 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 Z4, Def4; :: thesis: verum

end;

end;