reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:40, 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 `2 = d & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & 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 `2 = d & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & 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 `2 = d & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & 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 `2 = d
and
A5:
p3 `2 = c
and
A6:
p4 `2 = c
and
A7:
c <= p1 `2
and
A8:
p1 `2 <= d
and
A9:
a <= p2 `1
and
A10:
p2 `1 <= b
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:
( g2 is continuous & g2 is one-to-one )
by A1, A2, A19, Th53;
A23:
p4 `1 < b
by A12, A13, XXREAL_0:2;
then A24:
- 1 < (g2 . I) `1
by A1, A11, A17, Th64;
A25:
(g . I) `2 = c
by A6, A17;
then A26:
- 1 <= (g2 . O) `1
by A1, A9, A10, A11, A16, A17, A23, Th64;
A27:
(g2 . O) `1 <= 1
by A1, A9, A10, A11, A16, A17, A23, A25, Th64;
d - c > 0
by A2, XREAL_1:50;
then A28:
2 / (d - c) > 0
by XREAL_1:139;
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) ;
A29:
( f2 is continuous & f2 is one-to-one )
by A1, A2, A18, Th53;
A30:
p3 `1 > a
by A11, A12, XXREAL_0:2;
then A31:
(f2 . I) `1 <= 1
by A1, A2, A5, A13, A15, Th61;
A32:
(f . I) `2 = c
by A5, A15;
then A33:
(f2 . O) `2 <= 1
by A1, A2, A7, A8, A13, A14, A15, A30, Th61;
set x = the Element of (rng f2) /\ (rng g2);
A34:
dom g = the carrier of I[01]
by FUNCT_2:def 1;
then A35:
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I
by A17, FUNCT_1:13;
A36:
(g2 . I) `2 = - 1
by A2, A6, A17, Th57;
A37:
(g2 . O) `2 = 1
by A2, A4, A16, Th55;
b - a > 0
by A1, XREAL_1:50;
then A38:
2 / (b - a) > 0
by XREAL_1:139;
then
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism
by A28, Th50;
then A39:
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one
by TOPS_2:def 5;
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;
A44:
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_1:13;
then A45:
(g2 . I) `1 < (f2 . I) `1
by A12, A38, A28, A35, Th50;
- 1 <= (f2 . O) `2
by A1, A2, A7, A8, A13, A14, A15, A30, A32, Th61;
then
rng f2 meets rng g2
by A29, A43, A42, A33, A31, A41, A22, A37, A36, A26, A27, A24, A40, A45, Th29, JGRAPH_6:79;
then A46:
(rng f2) /\ (rng g2) <> {}
by XBOOLE_0:def 7;
then
the Element of (rng f2) /\ (rng g2) in rng g2
by XBOOLE_0:def 4;
then consider z2 being set such that
A47:
z2 in dom g2
and
A48:
the Element of (rng f2) /\ (rng g2) = g2 . z2
by FUNCT_1:def 3;
A49:
the Element of (rng f2) /\ (rng g2) = (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . (g . z2)
by A34, A47, A48, FUNCT_1:13;
A50:
g . z2 in rng g
by A34, A47, FUNCT_1:def 3;
g . z2 in the carrier of (TOP-REAL 2)
by A47, FUNCT_2:5;
then A51:
g . z2 in dom (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))))
by FUNCT_2:def 1;
the Element of (rng f2) /\ (rng g2) in rng f2
by A46, XBOOLE_0:def 4;
then consider z1 being set such that
A52:
z1 in dom f2
and
A53:
the Element of (rng f2) /\ (rng g2) = f2 . z1
by FUNCT_1:def 3;
A54:
f . z1 in rng f
by A44, A52, FUNCT_1:def 3;
f . z1 in the carrier of (TOP-REAL 2)
by A52, FUNCT_2:5;
then A55:
f . z1 in dom (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))))
by FUNCT_2:def 1;
the Element of (rng f2) /\ (rng g2) = (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . (f . z1)
by A44, A52, A53, FUNCT_1:13;
then
f . z1 = g . z2
by A49, A55, A51, A39, FUNCT_1:def 4;
hence
rng f meets rng g
by A54, A50, XBOOLE_0:3; verum