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); :: 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 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & c <= 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 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & c <= 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 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & c <= 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 = b and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 < p3 `2 and
A10: p3 `2 <= d and
A11: c <= p4 `2 and
A12: p4 `2 <= d and
A13: f . 0 = p1 and
A14: f . 1 = p3 and
A15: g . 0 = p2 and
A16: g . 1 = p4 and
A17: ( f is continuous & f is one-to-one ) and
A18: ( g is continuous & g is one-to-one ) and
A19: rng f c= closed_inside_of_rectangle (a,b,c,d) and
A20: 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 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) ;
A21: p1 `2 < p3 `2 by A8, A9, XXREAL_0:2;
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) `2 by A2, A11, A12, A16, Th60;
A23: (g2 . I) `1 = 1 by A1, A6, A16, Th56;
A24: (g2 . O) `1 = - 1 by A1, A4, A15, Th54;
(f . I) `1 = a by A5, A14;
then A25: - 1 <= (f2 . O) `2 by A2, A7, A10, A13, A14, A21, Th58;
A26: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
set x = the Element of (rng f2) /\ (rng g2);
A27: dom f = 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))))) . p3 = f2 . I by A14, FUNCT_1:13;
d - c > 0 by A2, XREAL_1:50;
then A29: 2 / (d - c) > 0 by XREAL_1:139;
b - a > 0 by A1, XREAL_1:50;
then A30: 2 / (b - a) > 0 by XREAL_1:139;
then A31: ((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, A29, Th51;
(f . O) `1 = a by A3, A13;
then A32: (f2 . I) `2 <= 1 by A2, A7, A10, A13, A14, A21, Th58;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A30, A29, Th51;
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: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A35: (f2 . I) `1 = - 1 by A1, A5, A14, Th54;
A36: (f2 . O) `1 = - 1 by A1, A3, A13, Th54;
A37: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
A38: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A39: ((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, A30, A29, Th51;
A40: (g2 . I) `2 <= 1 by A2, A11, A12, A16, Th60;
A41: dom g = the carrier of I[01] by FUNCT_2:def 1;
then A42: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A13, A27, FUNCT_1:13;
then rng f2 meets rng g2 by A31, A39, A42, A28, A37, A36, A35, A25, A32, A34, A26, A24, A23, A22, A40, A38, Th16, 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 A41, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A41, A44, FUNCT_1:def 3;
g . z2 in the carrier of (TOP-REAL 2) by A44, FUNCT_2:5;
then A48: 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 A43, XBOOLE_0:def 4;
then consider z1 being set such that
A49: z1 in dom f2 and
A50: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def 3;
A51: f . z1 in rng f by A27, A49, FUNCT_1:def 3;
f . z1 in the carrier of (TOP-REAL 2) by A49, FUNCT_2:5;
then A52: 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 A27, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A33, FUNCT_1:def 4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; :: thesis: verum