now :: thesis: for y being object holds
( ( y in [.(- (sqrt 2)),(- 1).] implies ex x being object st
( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x ) ) & ( ex x being object st
( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x ) implies y in [.(- (sqrt 2)),(- 1).] ) )

let y be object ; :: thesis: ( ( y in [.(- (sqrt 2)),(- 1).] implies ex x being object st
( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x ) ) & ( ex x being object st
( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x ) implies y in [.(- (sqrt 2)),(- 1).] ) )
thus ( y in [.(- (sqrt 2)),(- 1).] implies ex x being object st
( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x ) ) :: thesis: ( ex x being object st
( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x ) implies y in [.(- (sqrt 2)),(- 1).] )

proof

[.((3 / 4) * PI),PI.] c= ].(PI / 2),PI.] by Lm6, XXREAL_2:def 12;
then A1: sec | [.((3 / 4) * PI),PI.] is continuous by Th38, FCONT_1:16;
assume A2: y in [.(- (sqrt 2)),(- 1).] ; :: thesis: ex x being object st
( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x )
then reconsider y1 = y as Real ;
A3: (3 / 4) * PI <= PI by Lm6, XXREAL_1:2;
y1 in [.(sec . ((3 / 4) * PI)),(sec . PI).] \/ [.(sec . PI),(sec . ((3 / 4) * PI)).] by A2, Th31, XBOOLE_0:def 3;
then consider x being Real such that
A4: ( x in [.((3 / 4) * PI),PI.] & y1 = sec . x ) by A3, A1, Lm14, Th2, FCONT_2:15, XBOOLE_1:1;
take x ; :: thesis: ( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x )
thus ( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x ) by A4, Lm30, FUNCT_1:49; :: thesis: verum

end;

thus ( ex x being object st
( x in dom (sec | [.((3 / 4) * PI),PI.]) & y = (sec | [.((3 / 4) * PI),PI.]) . x ) implies y in [.(- (sqrt 2)),(- 1).] ) :: thesis: verum

proof

given x being object such that A5: x in dom (sec | [.((3 / 4) * PI),PI.]) and
A6: y = (sec | [.((3 / 4) * PI),PI.]) . x ; :: thesis: y in [.(- (sqrt 2)),(- 1).]
reconsider x1 = x as Real by A5;
y = sec . x1 by A5, A6, Lm30, FUNCT_1:49;
hence y in [.(- (sqrt 2)),(- 1).] by A5, Lm30, Th34; :: thesis: verum

end;

end;

hence rng (sec | [.((3 / 4) * PI),PI.]) = [.(- (sqrt 2)),(- 1).] by FUNCT_1:def 3; :: thesis: verum