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); :: thesis: 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 `1 = a & p4 `1 = a & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 < p4 `2 & p4 `2 <= d & 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 ; :: thesis: for f, g being Function of I[01] ,(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `1 = a & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 < p4 `2 & p4 `2 <= d & 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); :: thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `1 = a & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 < p4 `2 & p4 `2 <= d & 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 `1 = a and
A6: p4 `1 = a and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 < p3 `2 and
A10: p3 `2 < p4 `2 and
A11: p4 `2 <= d and
A12: f . 0 = p1 and
A13: f . 1 = p3 and
A14: g . 0 = p2 and
A15: g . 1 = p4 and
A16: ( f is continuous & f is one-to-one ) and
A17: ( g is continuous & g is one-to-one ) and
A18: rng f c= closed_inside_of_rectangle a,b,c,d and
A19: rng g c= closed_inside_of_rectangle a,b,c,d ; :: thesis: 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) ;
A20: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A17, Th53;
A21: (g . O) `1 = a by A4, A14;
A22: c < p2 `2 by A7, A8, XXREAL_0:2;
p2 `2 < p4 `2 by A9, A10, XXREAL_0:2;
then A23: (g2 . I) `2 <= 1 by A2, A11, A14, A15, A22, A21, Th58;
A24: rng g2 c= closed_inside_of_rectangle (- 1),1,(- 1),1 by A1, A2, A19, Th52;
A25: (g2 . I) `1 = - 1 by A1, A6, A15, Th54;
A26: (g2 . O) `1 = - 1 by A1, A4, A14, Th54;
A27: dom g = the carrier of I[01] by FUNCT_2:def 1;
then A28: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p2 = g2 . O by A14, FUNCT_1:23;
A29: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p4 = g2 . I by A15, A27, FUNCT_1:23;
d - c > 0 by A2, XREAL_1:52;
then A30: 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) ;
A31: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A16, Th53;
A32: rng f2 c= closed_inside_of_rectangle (- 1),1,(- 1),1 by A1, A2, A18, Th52;
A33: (f2 . I) `1 = - 1 by A1, A5, A13, Th54;
A34: (f . I) `1 = a by A5, A13;
A35: p3 `2 < d by A10, A11, XXREAL_0:2;
p1 `2 < p3 `2 by A8, A9, XXREAL_0:2;
then A36: - 1 <= (f2 . O) `2 by A2, A7, A12, A13, A35, A34, Th58;
A37: (f2 . O) `1 = - 1 by A1, A3, A12, Th54;
consider x being Element of (rng f2) /\ (rng g2);
A38: dom f = the carrier of I[01] by FUNCT_2:def 1;
then A39: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3 = f2 . I by A13, FUNCT_1:23;
b - a > 0 by A1, XREAL_1:52;
then A40: 2 / (b - a) > 0 by XREAL_1:141;
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, A30, Th51;
A42: ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3) `2 < ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p4) `2 by A10, A40, A30, Th51;
A43: ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p2) `2 < ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3) `2 by A9, A40, A30, Th51;
(AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p1 = f2 . O by A12, A38, FUNCT_1:23;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle (- 1),1,(- 1),1 by A41, A43, A42, A28, A39, A29, A37, A33, A36, A26, A25, A23, Th14;
then rng f2 meets rng g2 by A31, A32, A20, A24, 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 A27, A45, A46, FUNCT_1:23;
AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))) is being_homeomorphism by A40, A30, Th51;
then A48: AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))) is one-to-one by TOPS_2:def 5;
x in rng f2 by A44, XBOOLE_0:def 4;
then consider z1 being set such that
A49: z1 in dom f2 and
A50: x = f2 . z1 by FUNCT_1:def 5;
A51: f . z1 in rng f by A38, A49, FUNCT_1:def 5;
A52: g . z2 in rng g by A27, A45, FUNCT_1:def 5;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:7;
then A53: 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 A49, FUNCT_2:7;
then A54: 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 A38, A49, A50, FUNCT_1:23;
then f . z1 = g . z2 by A47, A54, A53, A48, FUNCT_1:def 8;
hence rng f meets rng g by A51, A52, XBOOLE_0:3; :: thesis: verum