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 `1 = a & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `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 = a & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `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 = a & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `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 = a
and
A5:
p3 `2 = d
and
A6:
p4 `2 = c
and
A7:
c <= p1 `2
and
A8:
p1 `2 < p2 `2
and
A9:
p2 `2 <= d
and
A10:
a <= p3 `1
and
A11:
p3 `1 <= b
and
A12:
a < p4 `1
and
A13:
p4 `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 . I) `1
by A1, A2, A6, A12, A13, A17, Th61;
A23:
(g2 . I) `1 <= 1
by A1, A2, A6, A12, A13, A17, Th61;
A24:
( g2 is continuous & g2 is one-to-one )
by A1, A2, A19, Th53;
A25:
(g2 . I) `2 = - 1
by A2, A6, A17, Th57;
A26:
(g2 . O) `1 = - 1
by A1, A4, A16, Th54;
A27:
rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
by A1, A2, A21, Th52;
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:
- 1 <= (f2 . I) `1
by A1, A2, A5, A10, A11, A15, Th59;
A30:
rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
by A1, A2, A20, Th52;
A31:
(f2 . I) `2 = 1
by A2, A5, A15, Th55;
p1 `2 <= d
by A8, A9, XXREAL_0:2;
then A32:
- 1 <= (f2 . O) `2
by A1, A2, A3, A7, A14, Th59;
A33:
(f2 . O) `1 = - 1
by A1, A3, A14, Th54;
A34:
( f2 is continuous & f2 is one-to-one )
by A1, A2, A18, Th53;
A35:
(f2 . I) `1 <= 1
by A1, A2, A5, A10, A11, A15, Th59;
set x = the Element of (rng f2) /\ (rng g2);
A36:
dom g = the carrier of I[01]
by FUNCT_2:def 1;
then A37:
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O
by A16, FUNCT_1:13;
A38:
(g . I) `2 = c
by A6, A17;
c < p2 `2
by A7, A8, XXREAL_0:2;
then A39:
(g2 . O) `2 <= 1
by A1, A2, A9, A12, A13, A16, A17, A38, Th61;
b - a > 0
by A1, XREAL_1:50;
then A40:
2 / (b - a) > 0
by XREAL_1:139;
then A41:
((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1) `2 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `2
by A8, A28, Th51;
A42:
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))))) . p1 = f2 . O
by A14, FUNCT_1:13;
then
rng f2 meets rng g2
by A41, A37, A34, A33, A31, A32, A29, A35, A30, A24, A26, A25, A39, A22, A23, A27, Th20, JGRAPH_6:79;
then A43:
(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
A44:
z2 in dom g2
and
A45:
the Element of (rng f2) /\ (rng g2) = g2 . z2
by FUNCT_1:def 3;
A46:
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 A36, A44, A45, FUNCT_1:13;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism
by A40, A28, Th51;
then A47:
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one
by TOPS_2:def 5;
the Element of (rng f2) /\ (rng g2) in rng f2
by A43, XBOOLE_0:def 4;
then consider z1 being set such that
A48:
z1 in dom f2
and
A49:
the Element of (rng f2) /\ (rng g2) = f2 . z1
by FUNCT_1:def 3;
A50:
f . z1 in rng f
by A42, A48, FUNCT_1:def 3;
A51:
g . z2 in rng g
by A36, A44, FUNCT_1:def 3;
g . z2 in the carrier of (TOP-REAL 2)
by A44, FUNCT_2:5;
then A52:
g . z2 in dom (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))))
by FUNCT_2:def 1;
f . z1 in the carrier of (TOP-REAL 2)
by A48, FUNCT_2:5;
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;
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 A42, A48, A49, FUNCT_1:13;
then
f . z1 = g . z2
by A46, A53, A52, A47, FUNCT_1:def 4;
hence
rng f meets rng g
by A50, A51, XBOOLE_0:3; verum