let X be non empty set ; :: thesis: for A being set
for r being Real holds
( rng (chi (r,A,X)) c= {0,r} & chi (r,A,X) is without+infty & chi (r,A,X) is without-infty )
let A be set ; :: thesis: for r being Real holds
( rng (chi (r,A,X)) c= {0,r} & chi (r,A,X) is without+infty & chi (r,A,X) is without-infty )
let r be Real; :: thesis: ( rng (chi (r,A,X)) c= {0,r} & chi (r,A,X) is without+infty & chi (r,A,X) is without-infty )

now :: thesis: for y being object st y in rng (chi (r,A,X)) holds
y in {0,r}

let y be object ; :: thesis: ( y in rng (chi (r,A,X)) implies b₁ in {0,r} )
assume y in rng (chi (r,A,X)) ; :: thesis: b₁ in {0,r}
then consider x being object such that
A1: ( x in dom (chi (r,A,X)) & y = (chi (r,A,X)) . x ) by FUNCT_1:def 3;

per cases ( x in A or not x in A ) ;

suppose x in A ; :: thesis: b₁ in {0,r}

then (chi (r,A,X)) . x = r by A1, Def1;
hence y in {0,r} by A1, TARSKI:def 2; :: thesis: verum

end;

suppose not x in A ; :: thesis: b₁ in {0,r}

then (chi (r,A,X)) . x = 0 by A1, Def1;
hence y in {0,r} by A1, TARSKI:def 2; :: thesis: verum

end;

end;

end;

hence rng (chi (r,A,X)) c= {0,r} ; :: thesis: ( chi (r,A,X) is without+infty & chi (r,A,X) is without-infty )
( chi (A,X) is without+infty & chi (A,X) is without-infty ) by Th3;
then ( r (#) (chi (A,X)) is without+infty & r (#) (chi (A,X)) is without-infty ) ;
hence ( chi (r,A,X) is without+infty & chi (r,A,X) is without-infty ) by Th1; :: thesis: verum