let X be non empty set ; :: thesis: for f being PartFunc of X,ExtREAL holds
( ( for x being set st x in dom f holds
-infty < f . x ) iff f is without-infty )
let f be PartFunc of X,ExtREAL ; :: thesis: ( ( for x being set st x in dom f holds
-infty < f . x ) iff f is without-infty )

hereby :: thesis: ( f is without-infty implies for x being set st x in dom f holds
-infty < f . x )

assume A1: for x being set st x in dom f holds
-infty < f . x ; :: thesis: f is without-infty

now

let x be set ; :: thesis: -infty < f . b₁

per cases ( x in dom f or not x in dom f ) ;

suppose x in dom f ; :: thesis: -infty < f . b₁

hence -infty < f . x by A1; :: thesis: verum

end;

suppose not x in dom f ; :: thesis: -infty < f . b₁

hence -infty < f . x by FUNCT_1:def 4; :: thesis: verum

end;

end;

end;

hence f is without-infty by Def5; :: thesis: verum

end;

assume f is without-infty ; :: thesis: for x being set st x in dom f holds
-infty < f . x
hence for x being set st x in dom f holds
-infty < f . x by Def5; :: thesis: verum