reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:83, XXREAL_1:1;
let p1, p2, p3, p4 be Point of (TOP-REAL 2); for a, b, c, d being real number
for f, g being Function of I[01] ,(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & rng f c= closed_inside_of_rectangle a,b,c,d & rng g c= closed_inside_of_rectangle a,b,c,d holds
rng f meets rng g
let a, b, c, d be real number ; for f, g being Function of I[01] ,(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & rng f c= closed_inside_of_rectangle a,b,c,d & rng g c= closed_inside_of_rectangle a,b,c,d holds
rng f meets rng g
let f, g be Function of I[01] ,(TOP-REAL 2); ( a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & rng f c= closed_inside_of_rectangle a,b,c,d & rng g c= closed_inside_of_rectangle a,b,c,d implies rng f meets rng g )
assume that
A1:
a < b
and
A2:
c < d
and
A3:
p1 `1 = a
and
A4:
p2 `1 = b
and
A5:
p3 `2 = c
and
A6:
p4 `2 = c
and
A7:
c <= p1 `2
and
A8:
p1 `2 <= d
and
A9:
c <= p2 `2
and
A10:
p2 `2 <= d
and
A11:
a < p4 `1
and
A12:
p4 `1 < p3 `1
and
A13:
p3 `1 <= b
and
A14:
f . 0 = p1
and
A15:
f . 1 = p3
and
A16:
g . 0 = p2
and
A17:
g . 1 = p4
and
A18:
( f is continuous & f is one-to-one )
and
A19:
( g is continuous & g is one-to-one )
and
A20:
rng f c= closed_inside_of_rectangle a,b,c,d
and
A21:
rng g c= closed_inside_of_rectangle a,b,c,d
; rng f meets rng g
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
set h = AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)));
reconsider g2 = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) * g as Function of I[01] ,(TOP-REAL 2) ;
A22:
1 >= (g2 . O) `2
by A1, A2, A4, A9, A10, A16, Th66;
p4 `1 <= b
by A12, A13, XXREAL_0:2;
then A23:
(g2 . I) `1 > - 1
by A1, A2, A6, A11, A17, Th66;
d - c > 0
by A2, XREAL_1:52;
then A24:
2 / (d - c) > 0
by XREAL_1:141;
reconsider f2 = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) * f as Function of I[01] ,(TOP-REAL 2) ;
A25:
( f2 is continuous & f2 is one-to-one )
by A1, A2, A18, Th53;
A26:
p3 `1 > a
by A11, A12, XXREAL_0:2;
then A27:
(f2 . I) `1 <= 1
by A1, A2, A5, A13, A15, Th61;
A28:
(f . I) `2 = c
by A5, A15;
then A29:
(f2 . O) `2 <= 1
by A1, A2, A7, A8, A13, A14, A15, A26, Th61;
consider x being Element of (rng f2) /\ (rng g2);
A30:
dom g = the carrier of I[01]
by FUNCT_2:def 1;
then A31:
(AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p4 = g2 . I
by A17, FUNCT_1:23;
b - a > 0
by A1, XREAL_1:52;
then A32:
2 / (b - a) > 0
by XREAL_1:141;
then
AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))) is being_homeomorphism
by A24, Th50;
then A33:
AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))) is one-to-one
by TOPS_2:def 5;
A34:
dom f = the carrier of I[01]
by FUNCT_2:def 1;
then
(AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3 = f2 . I
by A15, FUNCT_1:23;
then A35:
(g2 . I) `1 < (f2 . I) `1
by A12, A32, A24, A31, Th50;
A36:
(g2 . O) `2 >= - 1
by A1, A2, A4, A9, A10, A16, Th66;
A37:
( g2 is continuous & g2 is one-to-one )
by A1, A2, A19, Th53;
A38:
(g2 . I) `2 = - 1
by A2, A6, A17, Th57;
A39:
(g2 . O) `1 = 1
by A1, A4, A16, Th56;
A40:
rng g2 c= closed_inside_of_rectangle (- 1),1,(- 1),1
by A1, A2, A21, Th52;
A41:
rng f2 c= closed_inside_of_rectangle (- 1),1,(- 1),1
by A1, A2, A20, Th52;
A42:
(f2 . I) `2 = - 1
by A2, A5, A15, Th57;
A43:
(f2 . O) `1 = - 1
by A1, A3, A14, Th54;
- 1 <= (f2 . O) `2
by A1, A2, A7, A8, A13, A14, A15, A26, A28, Th61;
then
rng f2 meets rng g2
by A25, A43, A42, A29, A27, A41, A37, A39, A38, A22, A36, A23, A40, A35, Th32, JGRAPH_6:89;
then A44:
(rng f2) /\ (rng g2) <> {}
by XBOOLE_0:def 7;
then
x in rng g2
by XBOOLE_0:def 4;
then consider z2 being set such that
A45:
z2 in dom g2
and
A46:
x = g2 . z2
by FUNCT_1:def 5;
A47:
x = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . (g . z2)
by A30, A45, A46, FUNCT_1:23;
A48:
g . z2 in rng g
by A30, A45, FUNCT_1:def 5;
g . z2 in the carrier of (TOP-REAL 2)
by A45, FUNCT_2:7;
then A49:
g . z2 in dom (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))
by FUNCT_2:def 1;
x in rng f2
by A44, XBOOLE_0:def 4;
then consider z1 being set such that
A50:
z1 in dom f2
and
A51:
x = f2 . z1
by FUNCT_1:def 5;
A52:
f . z1 in rng f
by A34, A50, FUNCT_1:def 5;
f . z1 in the carrier of (TOP-REAL 2)
by A50, FUNCT_2:7;
then A53:
f . z1 in dom (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))
by FUNCT_2:def 1;
x = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . (f . z1)
by A34, A50, A51, FUNCT_1:23;
then
f . z1 = g . z2
by A47, A53, A49, A33, FUNCT_1:def 8;
hence
rng f meets rng g
by A52, A48, XBOOLE_0:3; verum