now

let y be set ; :: thesis: ( ( y in [.1,(sqrt 2).] implies ex x being set st
( x in dom (cosec | [.(PI / 4),(PI / 2).]) & y = (cosec | [.(PI / 4),(PI / 2).]) . x ) ) & ( ex x being set st
( x in dom (cosec | [.(PI / 4),(PI / 2).]) & y = (cosec | [.(PI / 4),(PI / 2).]) . x ) implies y in [.1,(sqrt 2).] ) )
thus ( y in [.1,(sqrt 2).] implies ex x being set st
( x in dom (cosec | [.(PI / 4),(PI / 2).]) & y = (cosec | [.(PI / 4),(PI / 2).]) . x ) ) :: thesis: ( ex x being set st
( x in dom (cosec | [.(PI / 4),(PI / 2).]) & y = (cosec | [.(PI / 4),(PI / 2).]) . x ) implies y in [.1,(sqrt 2).] )

proof

assume A1: y in [.1,(sqrt 2).] ; :: thesis: ex x being set st
( x in dom (cosec | [.(PI / 4),(PI / 2).]) & y = (cosec | [.(PI / 4),(PI / 2).]) . x )
then reconsider y1 = y as Real ;
A2: PI / 4 <= PI / 2 by Lm12, XXREAL_1:2;
[.(PI / 4),(PI / 2).] c= ].0 ,(PI / 2).] by Lm12, XXREAL_2:def 12;
then A4: cosec | [.(PI / 4),(PI / 2).] is continuous by Th40, FCONT_1:17;
y1 in [.(cosec . (PI / 2)),(cosec . (PI / 4)).] \/ [.(cosec . (PI / 4)),(cosec . (PI / 2)).] by A1, Th32, XBOOLE_0:def 3;
then consider x being Real such that
A5: x in [.(PI / 4),(PI / 2).] and
A6: y1 = cosec . x by A2, A4, X8, FCONT_2:16;
take x ; :: thesis: ( x in dom (cosec | [.(PI / 4),(PI / 2).]) & y = (cosec | [.(PI / 4),(PI / 2).]) . x )
thus ( x in dom (cosec | [.(PI / 4),(PI / 2).]) & y = (cosec | [.(PI / 4),(PI / 2).]) . x ) by A5, A6, Lm16, FUNCT_1:72; :: thesis: verum

end;

thus ( ex x being set st
( x in dom (cosec | [.(PI / 4),(PI / 2).]) & y = (cosec | [.(PI / 4),(PI / 2).]) . x ) implies y in [.1,(sqrt 2).] ) :: thesis: verum

proof

given x being set such that A7: x in dom (cosec | [.(PI / 4),(PI / 2).]) and
A8: y = (cosec | [.(PI / 4),(PI / 2).]) . x ; :: thesis: y in [.1,(sqrt 2).]
reconsider x1 = x as Real by A7;
y = cosec . x1 by A7, A8, Lm16, FUNCT_1:72;
hence y in [.1,(sqrt 2).] by A7, Lm16, Th36; :: thesis: verum

end;

end;

hence rng (cosec | [.(PI / 4),(PI / 2).]) = [.1,(sqrt 2).] by FUNCT_1:def 5; :: thesis: verum