:: JGRAPH_7 semantic presentation
begin
theorem Th1: :: JGRAPH_7:1
for a, b, d being real number
for p being Point of (TOP-REAL 2) st a < b & p `2 = d & a <= p `1 & p `1 <= b holds
p in LSeg (|[a,d]|,|[b,d]|)
proof
let a, b, d be real number ; ::_thesis: for p being Point of (TOP-REAL 2) st a < b & p `2 = d & a <= p `1 & p `1 <= b holds
p in LSeg (|[a,d]|,|[b,d]|)
let p be Point of (TOP-REAL 2); ::_thesis: ( a < b & p `2 = d & a <= p `1 & p `1 <= b implies p in LSeg (|[a,d]|,|[b,d]|) )
assume that
A1: a < b and
A2: p `2 = d and
A3: a <= p `1 and
A4: p `1 <= b ; ::_thesis: p in LSeg (|[a,d]|,|[b,d]|)
reconsider w = ((p `1) - a) / (b - a) as Real ;
A5: b - a > 0 by A1, XREAL_1:50;
(p `1) - a <= b - a by A4, XREAL_1:9;
then w <= (b - a) / (b - a) by A5, XREAL_1:72;
then A6: w <= 1 by A5, XCMPLX_1:60;
(p `1) - a >= 0 by A3, XREAL_1:48;
then A7: 0 <= w by A5, XREAL_1:136;
((1 - w) * |[a,d]|) + (w * |[b,d]|) = |[((1 - w) * a),((1 - w) * d)]| + (w * |[b,d]|) by EUCLID:58
.= |[((1 - w) * a),((1 - w) * d)]| + |[(w * b),(w * d)]| by EUCLID:58
.= |[(((1 - w) * a) + (w * b)),(((1 - w) * d) + (w * d))]| by EUCLID:56
.= |[(a + (w * (b - a))),d]|
.= |[(a + ((p `1) - a)),d]| by A5, XCMPLX_1:87
.= p by A2, EUCLID:53 ;
hence p in LSeg (|[a,d]|,|[b,d]|) by A7, A6; ::_thesis: verum
end;
theorem Th2: :: JGRAPH_7:2
for n being Element of NAT
for P being Subset of (TOP-REAL n)
for p1, p2 being Point of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 )
proof
let n be Element of NAT ; ::_thesis: for P being Subset of (TOP-REAL n)
for p1, p2 being Point of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 )
let P be Subset of (TOP-REAL n); ::_thesis: for p1, p2 being Point of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 )
let p1, p2 be Point of (TOP-REAL n); ::_thesis: ( P is_an_arc_of p1,p2 implies ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 ) )
assume A1: P is_an_arc_of p1,p2 ; ::_thesis: ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 )
then consider f2 being Function of I[01],((TOP-REAL n) | P) such that
A2: f2 is being_homeomorphism and
A3: f2 . 0 = p1 and
A4: f2 . 1 = p2 by TOPREAL1:def_1;
p1 in P by A1, TOPREAL1:1;
then consider g being Function of I[01],(TOP-REAL n) such that
A5: f2 = g and
A6: g is continuous and
A7: g is one-to-one by A2, JORDAN7:15;
rng g = [#] ((TOP-REAL n) | P) by A2, A5, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
hence ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 ) by A3, A4, A5, A6, A7; ::_thesis: verum
end;
theorem Th3: :: JGRAPH_7:3
for p1, p2 being Point of (TOP-REAL 2)
for b, c, d being real number st p1 `1 < b & p1 `1 = p2 `1 & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle ((p1 `1),b,c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for b, c, d being real number st p1 `1 < b & p1 `1 = p2 `1 & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle ((p1 `1),b,c,d)
let b, c, d be real number ; ::_thesis: ( p1 `1 < b & p1 `1 = p2 `1 & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d implies LE p1,p2, rectangle ((p1 `1),b,c,d) )
set a = p1 `1 ;
assume that
A1: p1 `1 < b and
A2: p1 `1 = p2 `1 and
A3: c <= p1 `2 and
A4: p1 `2 < p2 `2 and
A5: p2 `2 <= d ; ::_thesis: LE p1,p2, rectangle ((p1 `1),b,c,d)
A6: p1 `2 < d by A4, A5, XXREAL_0:2;
then A7: c < d by A3, XXREAL_0:2;
then A8: p1 in LSeg (|[(p1 `1),c]|,|[(p1 `1),d]|) by A3, A6, JGRAPH_6:2;
c <= p2 `2 by A3, A4, XXREAL_0:2;
then p2 in LSeg (|[(p1 `1),c]|,|[(p1 `1),d]|) by A2, A5, A7, JGRAPH_6:2;
hence LE p1,p2, rectangle ((p1 `1),b,c,d) by A1, A4, A7, A8, JGRAPH_6:55; ::_thesis: verum
end;
theorem Th4: :: JGRAPH_7:4
for p1, p2 being Point of (TOP-REAL 2)
for b, c being real number st p1 `1 < b & c < p2 `2 & c <= p1 `2 & p1 `2 <= p2 `2 & p1 `1 <= p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle ((p1 `1),b,c,(p2 `2))
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for b, c being real number st p1 `1 < b & c < p2 `2 & c <= p1 `2 & p1 `2 <= p2 `2 & p1 `1 <= p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle ((p1 `1),b,c,(p2 `2))
let b, c be real number ; ::_thesis: ( p1 `1 < b & c < p2 `2 & c <= p1 `2 & p1 `2 <= p2 `2 & p1 `1 <= p2 `1 & p2 `1 <= b implies LE p1,p2, rectangle ((p1 `1),b,c,(p2 `2)) )
set a = p1 `1 ;
set d = p2 `2 ;
assume that
A1: p1 `1 < b and
A2: c < p2 `2 and
A3: c <= p1 `2 and
A4: p1 `2 <= p2 `2 and
A5: p1 `1 <= p2 `1 and
A6: p2 `1 <= b ; ::_thesis: LE p1,p2, rectangle ((p1 `1),b,c,(p2 `2))
A7: p1 in LSeg (|[(p1 `1),c]|,|[(p1 `1),(p2 `2)]|) by A2, A3, A4, JGRAPH_6:2;
p2 in LSeg (|[(p1 `1),(p2 `2)]|,|[b,(p2 `2)]|) by A1, A5, A6, Th1;
hence LE p1,p2, rectangle ((p1 `1),b,c,(p2 `2)) by A1, A2, A7, JGRAPH_6:59; ::_thesis: verum
end;
theorem Th5: :: JGRAPH_7:5
for p1, p2 being Point of (TOP-REAL 2)
for c, d being real number st p1 `1 < p2 `1 & c < d & c <= p1 `2 & p1 `2 <= d & c <= p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle ((p1 `1),(p2 `1),c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for c, d being real number st p1 `1 < p2 `1 & c < d & c <= p1 `2 & p1 `2 <= d & c <= p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle ((p1 `1),(p2 `1),c,d)
let c, d be real number ; ::_thesis: ( p1 `1 < p2 `1 & c < d & c <= p1 `2 & p1 `2 <= d & c <= p2 `2 & p2 `2 <= d implies LE p1,p2, rectangle ((p1 `1),(p2 `1),c,d) )
set a = p1 `1 ;
set b = p2 `1 ;
assume that
A1: p1 `1 < p2 `1 and
A2: c < d and
A3: c <= p1 `2 and
A4: p1 `2 <= d and
A5: c <= p2 `2 and
A6: p2 `2 <= d ; ::_thesis: LE p1,p2, rectangle ((p1 `1),(p2 `1),c,d)
A7: p2 in LSeg (|[(p2 `1),c]|,|[(p2 `1),d]|) by A2, A5, A6, JGRAPH_6:2;
p1 in LSeg (|[(p1 `1),c]|,|[(p1 `1),d]|) by A2, A3, A4, JGRAPH_6:2;
hence LE p1,p2, rectangle ((p1 `1),(p2 `1),c,d) by A1, A2, A7, JGRAPH_6:59; ::_thesis: verum
end;
theorem Th6: :: JGRAPH_7:6
for p1, p2 being Point of (TOP-REAL 2)
for b, d being real number st p2 `2 < d & p2 `2 <= p1 `2 & p1 `2 <= d & p1 `1 < p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle ((p1 `1),b,(p2 `2),d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for b, d being real number st p2 `2 < d & p2 `2 <= p1 `2 & p1 `2 <= d & p1 `1 < p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle ((p1 `1),b,(p2 `2),d)
let b, d be real number ; ::_thesis: ( p2 `2 < d & p2 `2 <= p1 `2 & p1 `2 <= d & p1 `1 < p2 `1 & p2 `1 <= b implies LE p1,p2, rectangle ((p1 `1),b,(p2 `2),d) )
set a = p1 `1 ;
set c = p2 `2 ;
set K = rectangle ((p1 `1),b,(p2 `2),d);
assume that
A1: p2 `2 < d and
A2: p2 `2 <= p1 `2 and
A3: p1 `2 <= d and
A4: p1 `1 < p2 `1 and
A5: p2 `1 <= b ; ::_thesis: LE p1,p2, rectangle ((p1 `1),b,(p2 `2),d)
A6: p1 in LSeg (|[(p1 `1),(p2 `2)]|,|[(p1 `1),d]|) by A1, A2, A3, JGRAPH_6:2;
A7: p1 `1 < b by A4, A5, XXREAL_0:2;
then W-min (rectangle ((p1 `1),b,(p2 `2),d)) = |[(p1 `1),(p2 `2)]| by A1, JGRAPH_6:46;
then A8: (W-min (rectangle ((p1 `1),b,(p2 `2),d))) `1 = p1 `1 by EUCLID:52;
p2 in LSeg (|[b,(p2 `2)]|,|[(p1 `1),(p2 `2)]|) by A4, A5, A7, Th1;
hence LE p1,p2, rectangle ((p1 `1),b,(p2 `2),d) by A1, A4, A7, A6, A8, JGRAPH_6:59; ::_thesis: verum
end;
theorem Th7: :: JGRAPH_7:7
for p1, p2 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle (a,b,c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b implies LE p1,p2, rectangle (a,b,c,d) )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: a <= p1 `1 and
A6: p1 `1 < p2 `1 and
A7: p2 `1 <= b ; ::_thesis: LE p1,p2, rectangle (a,b,c,d)
a <= p2 `1 by A5, A6, XXREAL_0:2;
then A8: p2 in LSeg (|[a,d]|,|[b,d]|) by A1, A4, A7, Th1;
p1 `1 <= b by A6, A7, XXREAL_0:2;
then p1 in LSeg (|[a,d]|,|[b,d]|) by A1, A3, A5, Th1;
hence LE p1,p2, rectangle (a,b,c,d) by A1, A2, A6, A8, JGRAPH_6:60; ::_thesis: verum
end;
theorem Th8: :: JGRAPH_7:8
for p1, p2 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle (a,b,c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d implies LE p1,p2, rectangle (a,b,c,d) )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `1 = b and
A5: a <= p1 `1 and
A6: p1 `1 <= b and
A7: c <= p2 `2 and
A8: p2 `2 <= d ; ::_thesis: LE p1,p2, rectangle (a,b,c,d)
A9: p2 in LSeg (|[b,d]|,|[b,c]|) by A2, A4, A7, A8, JGRAPH_6:2;
p1 in LSeg (|[a,d]|,|[b,d]|) by A1, A3, A5, A6, Th1;
hence LE p1,p2, rectangle (a,b,c,d) by A1, A2, A9, JGRAPH_6:60; ::_thesis: verum
end;
theorem Th9: :: JGRAPH_7:9
for p1, p2 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle (a,b,c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p2 `1 & p2 `1 <= b implies LE p1,p2, rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = c and
A5: a <= p1 `1 and
A6: p1 `1 <= b and
A7: a < p2 `1 and
A8: p2 `1 <= b ; ::_thesis: LE p1,p2, rectangle (a,b,c,d)
A9: p2 in LSeg (|[b,c]|,|[a,c]|) by A1, A4, A7, A8, Th1;
W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, JGRAPH_6:46;
then A10: (W-min (rectangle (a,b,c,d))) `1 = a by EUCLID:52;
p1 in LSeg (|[a,d]|,|[b,d]|) by A1, A3, A5, A6, Th1;
hence LE p1,p2, rectangle (a,b,c,d) by A1, A2, A7, A9, A10, JGRAPH_6:60; ::_thesis: verum
end;
theorem Th10: :: JGRAPH_7:10
for p1, p2 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `1 = b & c <= p2 `2 & p2 `2 < p1 `2 & p1 `2 <= d holds
LE p1,p2, rectangle (a,b,c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `1 = b & c <= p2 `2 & p2 `2 < p1 `2 & p1 `2 <= d holds
LE p1,p2, rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `1 = b & c <= p2 `2 & p2 `2 < p1 `2 & p1 `2 <= d implies LE p1,p2, rectangle (a,b,c,d) )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `1 = b and
A5: c <= p2 `2 and
A6: p2 `2 < p1 `2 and
A7: p1 `2 <= d ; ::_thesis: LE p1,p2, rectangle (a,b,c,d)
d >= p2 `2 by A6, A7, XXREAL_0:2;
then A8: p2 in LSeg (|[b,d]|,|[b,c]|) by A2, A4, A5, JGRAPH_6:2;
p1 `2 >= c by A5, A6, XXREAL_0:2;
then p1 in LSeg (|[b,d]|,|[b,c]|) by A2, A3, A7, JGRAPH_6:2;
hence LE p1,p2, rectangle (a,b,c,d) by A1, A2, A6, A8, JGRAPH_6:61; ::_thesis: verum
end;
theorem Th11: :: JGRAPH_7:11
for p1, p2 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle (a,b,c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p2 `1 & p2 `1 <= b holds
LE p1,p2, rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p2 `1 & p2 `1 <= b implies LE p1,p2, rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `2 = c and
A5: c <= p1 `2 and
A6: p1 `2 <= d and
A7: a < p2 `1 and
A8: p2 `1 <= b ; ::_thesis: LE p1,p2, rectangle (a,b,c,d)
A9: p1 in LSeg (|[b,d]|,|[b,c]|) by A2, A3, A5, A6, JGRAPH_6:2;
W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, JGRAPH_6:46;
then A10: (W-min (rectangle (a,b,c,d))) `1 = a by EUCLID:52;
p2 in LSeg (|[b,c]|,|[a,c]|) by A1, A4, A7, A8, Th1;
hence LE p1,p2, rectangle (a,b,c,d) by A1, A2, A7, A9, A10, JGRAPH_6:61; ::_thesis: verum
end;
theorem Th12: :: JGRAPH_7:12
for p1, p2 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = c & p2 `2 = c & a < p2 `1 & p2 `1 < p1 `1 & p1 `1 <= b holds
LE p1,p2, rectangle (a,b,c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = c & p2 `2 = c & a < p2 `1 & p2 `1 < p1 `1 & p1 `1 <= b holds
LE p1,p2, rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = c & p2 `2 = c & a < p2 `1 & p2 `1 < p1 `1 & p1 `1 <= b implies LE p1,p2, rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = c and
A4: p2 `2 = c and
A5: a < p2 `1 and
A6: p2 `1 < p1 `1 and
A7: p1 `1 <= b ; ::_thesis: LE p1,p2, rectangle (a,b,c,d)
b > p2 `1 by A6, A7, XXREAL_0:2;
then A8: p2 in LSeg (|[b,c]|,|[a,c]|) by A1, A4, A5, Th1;
W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, JGRAPH_6:46;
then A9: (W-min (rectangle (a,b,c,d))) `1 = a by EUCLID:52;
p1 `1 > a by A5, A6, XXREAL_0:2;
then p1 in LSeg (|[b,c]|,|[a,c]|) by A1, A3, A7, Th1;
hence LE p1,p2, rectangle (a,b,c,d) by A1, A2, A5, A6, A8, A9, JGRAPH_6:62; ::_thesis: verum
end;
theorem Th13: :: JGRAPH_7:13
for p1, p2 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle (a,b,c,d)
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d holds
LE p1,p2, rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d implies LE p1,p2, rectangle (a,b,c,d) )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `1 = b and
A5: a <= p1 `1 and
A6: p1 `1 <= b and
A7: c <= p2 `2 and
A8: p2 `2 <= d ; ::_thesis: LE p1,p2, rectangle (a,b,c,d)
A9: p2 in LSeg (|[b,d]|,|[b,c]|) by A2, A4, A7, A8, JGRAPH_6:2;
p1 in LSeg (|[a,d]|,|[b,d]|) by A1, A3, A5, A6, Th1;
hence LE p1,p2, rectangle (a,b,c,d) by A1, A2, A9, JGRAPH_6:60; ::_thesis: verum
end;
theorem Th14: :: JGRAPH_7:14
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & 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 implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: p1 `1 = a and
A3: p2 `1 = a and
A4: p3 `1 = a and
A5: p4 `1 = a and
A6: c <= p1 `2 and
A7: p1 `2 < p2 `2 and
A8: p2 `2 < p3 `2 and
A9: p3 `2 < p4 `2 and
A10: p4 `2 <= d ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A11: p3 `2 < d by A9, A10, XXREAL_0:2;
p2 `2 < p4 `2 by A8, A9, XXREAL_0:2;
then A12: p2 `2 < d by A10, XXREAL_0:2;
A13: c < p2 `2 by A6, A7, XXREAL_0:2;
then c < p3 `2 by A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A12, A13, A11, Th3;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th15: :: JGRAPH_7:15
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a <= p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a <= p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a <= p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
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 `2 = d and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 < p3 `2 and
A10: p3 `2 <= d and
A11: a <= p4 `1 and
A12: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p2 `2 < d by A9, A10, XXREAL_0:2;
A14: c < p2 `2 by A7, A8, XXREAL_0:2;
then c < p3 `2 by A9, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th3, Th4;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th16: :: JGRAPH_7:16
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_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 implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
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 ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p2 `2 <= d by A9, A10, XXREAL_0:2;
A14: c < p2 `2 by A7, A8, XXREAL_0:2;
then c < p3 `2 by A9, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th3, Th5;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th17: :: JGRAPH_7:17
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
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 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 < p3 `2 and
A10: p3 `2 <= d and
A11: a < p4 `1 and
A12: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p2 `2 < d by A9, A10, XXREAL_0:2;
A14: c < p2 `2 by A7, A8, XXREAL_0:2;
then c < p3 `2 by A9, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th3, Th6;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th18: :: JGRAPH_7:18
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
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 = d 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 < p4 `1 and
A12: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `1 < b by A11, A12, XXREAL_0:2;
c < p2 `2 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th4, Th7;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th19: :: JGRAPH_7:19
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
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 `1 = b 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: c <= p4 `2 and
A13: p4 `2 <= d ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
c < p2 `2 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th4, Th8;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th20: :: JGRAPH_7:20
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( 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 implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
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 ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
c < p2 `2 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th4, Th9;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th21: :: JGRAPH_7:21
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = a and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: c <= p4 `2 and
A11: p4 `2 < p3 `2 and
A12: p3 `2 <= d ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `2 > c by A10, A11, XXREAL_0:2;
c < p2 `2 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th5, Th10;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th22: :: JGRAPH_7:22
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = a and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: c <= p3 `2 and
A11: p3 `2 <= d and
A12: a < p4 `1 and
A13: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
c < p2 `2 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th5, Th11;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th23: :: JGRAPH_7:23
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = a and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: a < p4 `1 and
A11: p4 `1 < p3 `1 and
A12: p3 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: a < p3 `1 by A10, A11, XXREAL_0:2;
c < p2 `2 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th3, Th6, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th24: :: JGRAPH_7:24
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = d and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < p3 `1 and
A11: p3 `1 < p4 `1 and
A12: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `1 < b by A11, A12, XXREAL_0:2;
p2 `1 < p4 `1 by A10, A11, XXREAL_0:2;
then A14: p2 `1 < b by A12, XXREAL_0:2;
a < p3 `1 by A9, A10, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th4, Th7;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th25: :: JGRAPH_7:25
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < p3 `1 and
A11: p3 `1 <= b and
A12: c <= p4 `2 and
A13: p4 `2 <= d ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A14: a < p3 `1 by A9, A10, XXREAL_0:2;
p2 `1 < b by A10, A11, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th4, Th7, Th13;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th26: :: JGRAPH_7:26
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < p3 `1 and
A11: p3 `1 <= b and
A12: a < p4 `1 and
A13: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A14: a < p3 `1 by A9, A10, XXREAL_0:2;
p2 `1 < b by A10, A11, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th4, Th7, Th9;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th27: :: JGRAPH_7:27
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 <= b and
A11: c <= p4 `2 and
A12: p4 `2 < p3 `2 and
A13: p3 `2 <= d ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
c < p3 `2 by A11, A12, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th4, Th8, Th10;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th28: :: JGRAPH_7:28
for p1, p2, p3, p4 being Point of (TOP-REAL 2) st p1 `1 <> p3 `1 & p4 `2 <> p2 `2 & p4 `2 <= p1 `2 & p1 `2 <= p2 `2 & p1 `1 <= p2 `1 & p2 `1 <= p3 `1 & p4 `2 <= p3 `2 & p3 `2 <= p2 `2 & p1 `1 < p4 `1 & p4 `1 <= p3 `1 holds
p1,p2,p3,p4 are_in_this_order_on rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2))
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: ( p1 `1 <> p3 `1 & p4 `2 <> p2 `2 & p4 `2 <= p1 `2 & p1 `2 <= p2 `2 & p1 `1 <= p2 `1 & p2 `1 <= p3 `1 & p4 `2 <= p3 `2 & p3 `2 <= p2 `2 & p1 `1 < p4 `1 & p4 `1 <= p3 `1 implies p1,p2,p3,p4 are_in_this_order_on rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) )
set K = rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2));
assume that
A1: p1 `1 <> p3 `1 and
A2: p4 `2 <> p2 `2 and
A3: p4 `2 <= p1 `2 and
A4: p1 `2 <= p2 `2 and
A5: p1 `1 <= p2 `1 and
A6: p2 `1 <= p3 `1 and
A7: p4 `2 <= p3 `2 and
A8: p3 `2 <= p2 `2 and
A9: p1 `1 < p4 `1 and
A10: p4 `1 <= p3 `1 ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2))
p4 `2 <= p2 `2 by A3, A4, XXREAL_0:2;
then A11: p4 `2 < p2 `2 by A2, XXREAL_0:1;
p1 `1 <= p3 `1 by A5, A6, XXREAL_0:2;
then p1 `1 < p3 `1 by A1, XXREAL_0:1;
then ( ( LE p1,p2, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) & LE p2,p3, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) & LE p3,p4, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) ) or ( LE p2,p3, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) & LE p3,p4, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) & LE p4,p1, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) ) or ( LE p3,p4, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) & LE p4,p1, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) & LE p1,p2, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) ) or ( LE p4,p1, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) & LE p1,p2, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) & LE p2,p3, rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) ) ) by A3, A4, A5, A6, A7, A8, A9, A10, A11, Th4, Th8, Th11;
hence p1,p2,p3,p4 are_in_this_order_on rectangle ((p1 `1),(p3 `1),(p4 `2),(p2 `2)) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th29: :: JGRAPH_7:29
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( 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 implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
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 ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
a < p3 `1 by A11, A12, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th4, Th9, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th30: :: JGRAPH_7:30
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: d >= p2 `2 and
A10: p2 `2 > p3 `2 and
A11: p3 `2 > p4 `2 and
A12: p4 `2 >= c ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `2 < d by A9, A10, XXREAL_0:2;
p2 `2 > p4 `2 by A10, A11, XXREAL_0:2;
then A14: p2 `2 > c by A12, XXREAL_0:2;
c < p3 `2 by A11, A12, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th5, Th10;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th31: :: JGRAPH_7:31
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: d >= p2 `2 and
A10: p2 `2 > p3 `2 and
A11: p3 `2 >= c and
A12: a < p4 `1 and
A13: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A14: p3 `2 < d by A9, A10, XXREAL_0:2;
p2 `2 > c by A10, A11, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th5, Th10, Th11;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th32: :: JGRAPH_7:32
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number 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 holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( 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 implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
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 ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
p3 `1 > a by A11, A12, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th5, Th11, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th33: :: JGRAPH_7:33
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a < p4 `1 and
A10: p4 `1 < p3 `1 and
A11: p3 `1 < p2 `1 and
A12: p2 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `1 < b by A11, A12, XXREAL_0:2;
p2 `1 > p4 `1 by A10, A11, XXREAL_0:2;
then A14: p2 `1 > a by A9, XXREAL_0:2;
a < p3 `1 by A9, A10, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th6, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th34: :: JGRAPH_7:34
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = d and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 < p4 `1 and
A11: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A12: p3 `1 < b by A10, A11, XXREAL_0:2;
p2 `1 < p4 `1 by A9, A10, XXREAL_0:2;
then A13: p2 `1 < b by A11, XXREAL_0:2;
A14: a < p2 `1 by A7, A8, XXREAL_0:2;
then a < p3 `1 by A9, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A13, A14, A12, Th7;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th35: :: JGRAPH_7:35
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 <= b and
A11: c <= p4 `2 and
A12: p4 `2 <= d ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p2 `1 < b by A9, A10, XXREAL_0:2;
A14: a < p2 `1 by A7, A8, XXREAL_0:2;
then a < p3 `1 by A9, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th7, Th8;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th36: :: JGRAPH_7:36
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 <= b and
A11: a < p4 `1 and
A12: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: a < p2 `1 by A7, A8, XXREAL_0:2;
p3 `1 > p1 `1 by A8, A9, XXREAL_0:2;
then A14: p3 `1 > a by A7, XXREAL_0:2;
p2 `1 <= b by A9, A10, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th7, Th9;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th37: :: JGRAPH_7:37
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: c <= p4 `2 and
A11: p4 `2 < p3 `2 and
A12: p3 `2 <= d ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `2 > c by A10, A11, XXREAL_0:2;
a < p2 `1 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th7, Th8, Th10;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th38: :: JGRAPH_7:38
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: c <= p3 `2 and
A11: p3 `2 <= d and
A12: a < p4 `1 and
A13: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
a < p2 `1 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th7, Th8, Th11;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th39: :: JGRAPH_7:39
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: a < p4 `1 and
A11: p4 `1 < p3 `1 and
A12: p3 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `1 > a by A10, A11, XXREAL_0:2;
a < p2 `1 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th7, Th9, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th40: :: JGRAPH_7:40
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: d >= p2 `2 and
A10: p2 `2 > p3 `2 and
A11: p3 `2 > p4 `2 and
A12: p4 `2 >= c ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `2 > c by A11, A12, XXREAL_0:2;
p2 `2 > p4 `2 by A10, A11, XXREAL_0:2;
then A14: p2 `2 > c by A12, XXREAL_0:2;
d > p3 `2 by A9, A10, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th8, Th10;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th41: :: JGRAPH_7:41
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: d >= p2 `2 and
A10: p2 `2 > p3 `2 and
A11: p3 `2 >= c and
A12: a < p4 `1 and
A13: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A14: d > p3 `2 by A9, A10, XXREAL_0:2;
p2 `2 > c by A10, A11, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th8, Th10, Th11;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th42: :: JGRAPH_7:42
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `1 = b and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b 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 ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
a < p3 `1 by A11, A12, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th8, Th11, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th43: :: JGRAPH_7:43
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: a < p4 `1 and
A10: p4 `1 < p3 `1 and
A11: p3 `1 < p2 `1 and
A12: p2 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `1 < b by A11, A12, XXREAL_0:2;
p2 `1 > p4 `1 by A10, A11, XXREAL_0:2;
then A14: p2 `1 > a by A9, XXREAL_0:2;
a < p3 `1 by A9, A10, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th9, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th44: :: JGRAPH_7:44
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 > p3 `2 and
A10: p3 `2 > p4 `2 and
A11: p4 `2 >= c ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A12: p3 `2 > c by A10, A11, XXREAL_0:2;
p2 `2 > p4 `2 by A9, A10, XXREAL_0:2;
then A13: p2 `2 > c by A11, XXREAL_0:2;
A14: d > p2 `2 by A7, A8, XXREAL_0:2;
then d > p3 `2 by A9, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A13, A14, A12, Th10;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th45: :: JGRAPH_7:45
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 > p3 `2 and
A10: p3 `2 >= c and
A11: a < p4 `1 and
A12: p4 `1 <= b ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p2 `2 > c by A9, A10, XXREAL_0:2;
A14: d > p2 `2 by A7, A8, XXREAL_0:2;
then d > p3 `2 by A9, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, Th10, Th11;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th46: :: JGRAPH_7:46
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `1 = b and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 >= c and
A10: b >= p3 `1 and
A11: p3 `1 > p4 `1 and
A12: p4 `1 > a ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `1 > a by A11, A12, XXREAL_0:2;
d > p2 `2 by A7, A8, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th10, Th11, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th47: :: JGRAPH_7:47
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `1 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: b >= p2 `1 and
A10: p2 `1 > p3 `1 and
A11: p3 `1 > p4 `1 and
A12: p4 `1 > a ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A13: p3 `1 > a by A11, A12, XXREAL_0:2;
p2 `1 > p4 `1 by A10, A11, XXREAL_0:2;
then A14: p2 `1 > a by A12, XXREAL_0:2;
b > p3 `1 by A9, A10, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A14, A13, Th11, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th48: :: JGRAPH_7:48
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number st a < b & c < d & p1 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number st a < b & c < d & p1 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a holds
p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d & p1 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a implies p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) )
set K = rectangle (a,b,c,d);
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = c and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: b >= p1 `1 and
A8: p1 `1 > p2 `1 and
A9: p2 `1 > p3 `1 and
A10: p3 `1 > p4 `1 and
A11: p4 `1 > a ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d)
A12: p3 `1 > a by A10, A11, XXREAL_0:2;
p2 `1 > p4 `1 by A9, A10, XXREAL_0:2;
then A13: p2 `1 > a by A11, XXREAL_0:2;
A14: b > p2 `1 by A7, A8, XXREAL_0:2;
then b > p3 `1 by A9, XXREAL_0:2;
then ( ( LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) ) or ( LE p2,p3, rectangle (a,b,c,d) & LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) ) or ( LE p3,p4, rectangle (a,b,c,d) & LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) ) or ( LE p4,p1, rectangle (a,b,c,d) & LE p1,p2, rectangle (a,b,c,d) & LE p2,p3, rectangle (a,b,c,d) ) ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A13, A14, A12, Th12;
hence p1,p2,p3,p4 are_in_this_order_on rectangle (a,b,c,d) by JORDAN17:def_1; ::_thesis: verum
end;
theorem Th49: :: JGRAPH_7:49
for A, B, C, D being real number
for h, g being Function of (TOP-REAL 2),(TOP-REAL 2) st A > 0 & C > 0 & h = AffineMap (A,B,C,D) & g = AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C))) holds
( g = h " & h = g " )
proof
let A, B, C, D be real number ; ::_thesis: for h, g being Function of (TOP-REAL 2),(TOP-REAL 2) st A > 0 & C > 0 & h = AffineMap (A,B,C,D) & g = AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C))) holds
( g = h " & h = g " )
let h, g be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( A > 0 & C > 0 & h = AffineMap (A,B,C,D) & g = AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C))) implies ( g = h " & h = g " ) )
assume that
A1: A > 0 and
A2: C > 0 and
A3: h = AffineMap (A,B,C,D) and
A4: g = AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C))) ; ::_thesis: ( g = h " & h = g " )
A5: h is one-to-one by A1, A2, A3, JGRAPH_2:44;
A6: for x, y being set st x in dom h & y in dom g holds
( h . x = y iff g . y = x )
proof
let x, y be set ; ::_thesis: ( x in dom h & y in dom g implies ( h . x = y iff g . y = x ) )
assume that
A7: x in dom h and
A8: y in dom g ; ::_thesis: ( h . x = y iff g . y = x )
reconsider py = y as Point of (TOP-REAL 2) by A8;
reconsider px = x as Point of (TOP-REAL 2) by A7;
A9: ( h . x = y implies g . y = x )
proof
assume A10: h . x = y ; ::_thesis: g . y = x
A11: h . px = |[((A * (px `1)) + B),((C * (px `2)) + D)]| by A3, JGRAPH_2:def_2;
then py `1 = (A * (px `1)) + B by A10, EUCLID:52;
then A12: ((1 / A) * (py `1)) + (- (B / A)) = ((((1 / A) * A) * (px `1)) + ((1 / A) * B)) + (- (B / A))
.= ((1 * (px `1)) + ((1 / A) * B)) + (- (B / A)) by A1, XCMPLX_1:106
.= ((px `1) + (B / A)) + (- (B / A)) by XCMPLX_1:99
.= px `1 ;
py `2 = (C * (px `2)) + D by A10, A11, EUCLID:52;
then A13: ((1 / C) * (py `2)) + (- (D / C)) = ((((1 / C) * C) * (px `2)) + ((1 / C) * D)) + (- (D / C))
.= ((1 * (px `2)) + ((1 / C) * D)) + (- (D / C)) by A2, XCMPLX_1:106
.= ((px `2) + (D / C)) + (- (D / C)) by XCMPLX_1:99
.= px `2 ;
g . py = |[(((1 / A) * (py `1)) + (- (B / A))),(((1 / C) * (py `2)) + (- (D / C)))]| by A4, JGRAPH_2:def_2;
hence g . y = x by A12, A13, EUCLID:53; ::_thesis: verum
end;
( g . y = x implies h . x = y )
proof
assume A14: g . y = x ; ::_thesis: h . x = y
A15: g . py = |[(((1 / A) * (py `1)) + (- (B / A))),(((1 / C) * (py `2)) + (- (D / C)))]| by A4, JGRAPH_2:def_2;
then px `1 = ((1 / A) * (py `1)) + (- (B / A)) by A14, EUCLID:52;
then A16: (A * (px `1)) + B = (((A * (1 / A)) * (py `1)) + (A * (- (B / A)))) + B
.= ((1 * (py `1)) + (A * (- (B / A)))) + B by A1, XCMPLX_1:106
.= ((py `1) + (A * ((- B) / A))) + B by XCMPLX_1:187
.= ((py `1) + (- B)) + B by A1, XCMPLX_1:87
.= py `1 ;
px `2 = ((1 / C) * (py `2)) + (- (D / C)) by A14, A15, EUCLID:52;
then A17: (C * (px `2)) + D = (((C * (1 / C)) * (py `2)) + (C * (- (D / C)))) + D
.= ((1 * (py `2)) + (C * (- (D / C)))) + D by A2, XCMPLX_1:106
.= ((py `2) + (C * ((- D) / C))) + D by XCMPLX_1:187
.= ((py `2) + (- D)) + D by A2, XCMPLX_1:87
.= py `2 ;
h . px = |[((A * (px `1)) + B),((C * (px `2)) + D)]| by A3, JGRAPH_2:def_2;
hence h . x = y by A16, A17, EUCLID:53; ::_thesis: verum
end;
hence ( h . x = y iff g . y = x ) by A9; ::_thesis: verum
end;
A18: dom g = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
reconsider RD = D as Real by XREAL_0:def_1;
reconsider RC = C as Real by XREAL_0:def_1;
reconsider RB = B as Real by XREAL_0:def_1;
reconsider RA = A as Real by XREAL_0:def_1;
A19: g = AffineMap ((1 / RA),(- (RB / RA)),(1 / RC),(- (RD / RC))) by A4;
h = AffineMap (RA,RB,RC,RD) by A3;
then h is onto by A1, A2, JORDAN1K:36;
then A20: rng h = the carrier of (TOP-REAL 2) by FUNCT_2:def_3;
A21: 1 / C > 0 by A2, XREAL_1:139;
1 / A > 0 by A1, XREAL_1:139;
then g is onto by A21, A19, JORDAN1K:36;
then A22: rng g = the carrier of (TOP-REAL 2) by FUNCT_2:def_3;
dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then g = h " by A5, A18, A20, A22, A6, FUNCT_1:38;
hence ( g = h " & h = g " ) by A5, FUNCT_1:43; ::_thesis: verum
end;
theorem Th50: :: JGRAPH_7:50
for A, B, C, D being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2) st A > 0 & C > 0 & h = AffineMap (A,B,C,D) holds
( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `1 < p2 `1 holds
(h . p1) `1 < (h . p2) `1 ) )
proof
let A, B, C, D be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2) st A > 0 & C > 0 & h = AffineMap (A,B,C,D) holds
( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `1 < p2 `1 holds
(h . p1) `1 < (h . p2) `1 ) )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( A > 0 & C > 0 & h = AffineMap (A,B,C,D) implies ( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `1 < p2 `1 holds
(h . p1) `1 < (h . p2) `1 ) ) )
assume that
A1: A > 0 and
A2: C > 0 and
A3: h = AffineMap (A,B,C,D) ; ::_thesis: ( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `1 < p2 `1 holds
(h . p1) `1 < (h . p2) `1 ) )
A4: h is one-to-one by A1, A2, A3, JGRAPH_2:44;
set g = AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C)));
A5: AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C))) = h " by A1, A2, A3, Th49;
A6: for p1, p2 being Point of (TOP-REAL 2) st p1 `1 < p2 `1 holds
(h . p1) `1 < (h . p2) `1
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 `1 < p2 `1 implies (h . p1) `1 < (h . p2) `1 )
h . p1 = |[((A * (p1 `1)) + B),((C * (p1 `2)) + D)]| by A3, JGRAPH_2:def_2;
then A7: (h . p1) `1 = (A * (p1 `1)) + B by EUCLID:52;
h . p2 = |[((A * (p2 `1)) + B),((C * (p2 `2)) + D)]| by A3, JGRAPH_2:def_2;
then A8: (h . p2) `1 = (A * (p2 `1)) + B by EUCLID:52;
assume p1 `1 < p2 `1 ; ::_thesis: (h . p1) `1 < (h . p2) `1
then A * (p1 `1) < A * (p2 `1) by A1, XREAL_1:68;
hence (h . p1) `1 < (h . p2) `1 by A7, A8, XREAL_1:8; ::_thesis: verum
end;
A9: dom h = [#] (TOP-REAL 2) by FUNCT_2:def_1;
dom (AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C)))) = [#] (TOP-REAL 2) by FUNCT_2:def_1;
then A10: rng h = [#] (TOP-REAL 2) by A4, A5, FUNCT_1:32;
then ( h is onto & h is one-to-one ) by A1, A2, A3, FUNCT_2:def_3, JGRAPH_2:44;
then h /" is continuous by A5, TOPS_2:def_4;
hence ( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `1 < p2 `1 holds
(h . p1) `1 < (h . p2) `1 ) ) by A3, A4, A9, A10, A6, TOPS_2:def_5; ::_thesis: verum
end;
theorem Th51: :: JGRAPH_7:51
for A, B, C, D being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2) st A > 0 & C > 0 & h = AffineMap (A,B,C,D) holds
( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
(h . p1) `2 < (h . p2) `2 ) )
proof
let A, B, C, D be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2) st A > 0 & C > 0 & h = AffineMap (A,B,C,D) holds
( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
(h . p1) `2 < (h . p2) `2 ) )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( A > 0 & C > 0 & h = AffineMap (A,B,C,D) implies ( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
(h . p1) `2 < (h . p2) `2 ) ) )
assume that
A1: A > 0 and
A2: C > 0 and
A3: h = AffineMap (A,B,C,D) ; ::_thesis: ( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
(h . p1) `2 < (h . p2) `2 ) )
A4: h is one-to-one by A1, A2, A3, JGRAPH_2:44;
set g = AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C)));
A5: AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C))) = h " by A1, A2, A3, Th49;
A6: for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
(h . p1) `2 < (h . p2) `2
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 `2 < p2 `2 implies (h . p1) `2 < (h . p2) `2 )
h . p1 = |[((A * (p1 `1)) + B),((C * (p1 `2)) + D)]| by A3, JGRAPH_2:def_2;
then A7: (h . p1) `2 = (C * (p1 `2)) + D by EUCLID:52;
h . p2 = |[((A * (p2 `1)) + B),((C * (p2 `2)) + D)]| by A3, JGRAPH_2:def_2;
then A8: (h . p2) `2 = (C * (p2 `2)) + D by EUCLID:52;
assume p1 `2 < p2 `2 ; ::_thesis: (h . p1) `2 < (h . p2) `2
then C * (p1 `2) < C * (p2 `2) by A2, XREAL_1:68;
hence (h . p1) `2 < (h . p2) `2 by A7, A8, XREAL_1:8; ::_thesis: verum
end;
A9: dom h = [#] (TOP-REAL 2) by FUNCT_2:def_1;
dom (AffineMap ((1 / A),(- (B / A)),(1 / C),(- (D / C)))) = [#] (TOP-REAL 2) by FUNCT_2:def_1;
then A10: rng h = [#] (TOP-REAL 2) by A4, A5, FUNCT_1:32;
then ( h is onto & h is one-to-one ) by A1, A2, A3, FUNCT_2:def_3, JGRAPH_2:44;
then h /" is continuous by A5, TOPS_2:def_4;
hence ( h is being_homeomorphism & ( for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
(h . p1) `2 < (h . p2) `2 ) ) by A3, A4, A9, A10, A6, TOPS_2:def_5; ::_thesis: verum
end;
theorem Th52: :: JGRAPH_7:52
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2) st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & rng f c= closed_inside_of_rectangle (a,b,c,d) holds
rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2) st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & rng f c= closed_inside_of_rectangle (a,b,c,d) holds
rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2) st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & rng f c= closed_inside_of_rectangle (a,b,c,d) holds
rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
let f be Function of I[01],(TOP-REAL 2); ::_thesis: ( a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & rng f c= closed_inside_of_rectangle (a,b,c,d) implies rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: c < d and
A3: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A4: rng f c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (h * f) or x in closed_inside_of_rectangle ((- 1),1,(- 1),1) )
assume x in rng (h * f) ; ::_thesis: x in closed_inside_of_rectangle ((- 1),1,(- 1),1)
then consider y being set such that
A5: y in dom (h * f) and
A6: x = (h * f) . y by FUNCT_1:def_3;
reconsider t0 = y as Point of I[01] by A5;
A7: (h * f) . t0 = h . (f . t0) by A5, FUNCT_1:12;
dom f = the carrier of I[01] by FUNCT_2:def_1;
then f . t0 in rng f by FUNCT_1:def_3;
then f . t0 in closed_inside_of_rectangle (a,b,c,d) by A4;
then f . t0 in { p where p is Point of (TOP-REAL 2) : ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) } by JGRAPH_6:def_2;
then A8: ex p being Point of (TOP-REAL 2) st
( f . t0 = p & a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) ;
reconsider p0 = x as Point of (TOP-REAL 2) by A5, A6, FUNCT_2:5;
A9: h . (f . t0) = |[(((2 / (b - a)) * ((f . t0) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . t0) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
A10: b - a > 0 by A1, XREAL_1:50;
then A11: 2 / (b - a) > 0 by XREAL_1:139;
((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a)) = ((- 1) + ((b + a) / (b - a))) / (2 / (b - a))
.= ((((- 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A10, XCMPLX_1:113
.= (((a + a) / (b - a)) / 2) * (b - a) by XCMPLX_1:82
.= ((b - a) * ((a + a) / (b - a))) / 2
.= (a + a) / 2 by A10, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) <= (2 / (b - a)) * ((f . t0) `1) by A11, A8, XREAL_1:64;
then (- 1) - (- ((b + a) / (b - a))) <= (2 / (b - a)) * ((f . t0) `1) by A11, XCMPLX_1:87;
then ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) <= ((2 / (b - a)) * ((f . t0) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
then A12: - 1 <= p0 `1 by A6, A9, A7, EUCLID:52;
A13: d - c > 0 by A2, XREAL_1:50;
then A14: 2 / (d - c) > 0 by XREAL_1:139;
(1 - (- ((b + a) / (b - a)))) / (2 / (b - a)) = (1 + ((b + a) / (b - a))) / (2 / (b - a))
.= (((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A10, XCMPLX_1:113
.= (((b + b) / (b - a)) / 2) * (b - a) by XCMPLX_1:82
.= ((b - a) * ((b + b) / (b - a))) / 2
.= (b + b) / 2 by A10, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . t0) `1) by A11, A8, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . t0) `1) by A11, XCMPLX_1:87;
then (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . t0) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
then A15: p0 `1 <= 1 by A6, A9, A7, EUCLID:52;
(1 - (- ((d + c) / (d - c)))) / (2 / (d - c)) = (1 + ((d + c) / (d - c))) / (2 / (d - c))
.= (((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A13, XCMPLX_1:113
.= (((d + d) / (d - c)) / 2) * (d - c) by XCMPLX_1:82
.= ((d - c) * ((d + d) / (d - c))) / 2
.= (d + d) / 2 by A13, XCMPLX_1:87
.= d ;
then (2 / (d - c)) * ((1 - (- ((d + c) / (d - c)))) / (2 / (d - c))) >= (2 / (d - c)) * ((f . t0) `2) by A14, A8, XREAL_1:64;
then 1 - (- ((d + c) / (d - c))) >= (2 / (d - c)) * ((f . t0) `2) by A14, XCMPLX_1:87;
then (1 - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . t0) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
then A16: p0 `2 <= 1 by A6, A9, A7, EUCLID:52;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A13, XCMPLX_1:113
.= (((c + c) / (d - c)) / 2) * (d - c) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A13, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . t0) `2) by A14, A8, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . t0) `2) by A14, XCMPLX_1:87;
then ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . t0) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
then - 1 <= p0 `2 by A6, A9, A7, EUCLID:52;
then x in { p2 where p2 is Point of (TOP-REAL 2) : ( - 1 <= p2 `1 & p2 `1 <= 1 & - 1 <= p2 `2 & p2 `2 <= 1 ) } by A16, A12, A15;
hence x in closed_inside_of_rectangle ((- 1),1,(- 1),1) by JGRAPH_6:def_2; ::_thesis: verum
end;
theorem Th53: :: JGRAPH_7:53
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2) st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & f is continuous & f is one-to-one holds
( h * f is continuous & h * f is one-to-one )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2) st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & f is continuous & f is one-to-one holds
( h * f is continuous & h * f is one-to-one )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2) st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & f is continuous & f is one-to-one holds
( h * f is continuous & h * f is one-to-one )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: ( a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & f is continuous & f is one-to-one implies ( h * f is continuous & h * f is one-to-one ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: c < d and
A3: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A4: ( f is continuous & f is one-to-one ) ; ::_thesis: ( h * f is continuous & h * f is one-to-one )
d - c > 0 by A2, XREAL_1:50;
then A5: 2 / (d - c) > 0 by XREAL_1:139;
b - a > 0 by A1, XREAL_1:50;
then 2 / (b - a) > 0 by XREAL_1:139;
then h is being_homeomorphism by A3, A5, Th51;
then h is one-to-one by TOPS_2:def_5;
hence ( h * f is continuous & h * f is one-to-one ) by A3, A4, FUNCT_1:24; ::_thesis: verum
end;
theorem Th54: :: JGRAPH_7:54
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . O) `1 = a holds
((h * f) . O) `1 = - 1
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . O) `1 = a holds
((h * f) . O) `1 = - 1
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . O) `1 = a holds
((h * f) . O) `1 = - 1
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . O) `1 = a holds
((h * f) . O) `1 = - 1
let O be Point of I[01]; ::_thesis: ( a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . O) `1 = a implies ((h * f) . O) `1 = - 1 )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: (f . O) `1 = a ; ::_thesis: ((h * f) . O) `1 = - 1
A4: b - a > 0 by A1, XREAL_1:50;
dom f = the carrier of I[01] by FUNCT_2:def_1;
then A5: (h * f) . O = h . (f . O) by FUNCT_1:13;
A6: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) = ((2 * a) / (b - a)) + (- ((b + a) / (b - a))) by A3, XCMPLX_1:74
.= ((2 * a) / (b - a)) + ((- (b + a)) / (b - a)) by XCMPLX_1:187
.= ((2 * a) + (- (b + a))) / (b - a) by XCMPLX_1:62
.= (- (b - a)) / (b - a)
.= - 1 by A4, XCMPLX_1:197 ;
hence ((h * f) . O) `1 = - 1 by A5, A6, EUCLID:52; ::_thesis: verum
end;
theorem Th55: :: JGRAPH_7:55
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = d holds
((h * f) . I) `2 = 1
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = d holds
((h * f) . I) `2 = 1
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = d holds
((h * f) . I) `2 = 1
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = d holds
((h * f) . I) `2 = 1
let I be Point of I[01]; ::_thesis: ( c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = d implies ((h * f) . I) `2 = 1 )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: c < d and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: (f . I) `2 = d ; ::_thesis: ((h * f) . I) `2 = 1
A4: d - c > 0 by A1, XREAL_1:50;
dom f = the carrier of I[01] by FUNCT_2:def_1;
then A5: (h * f) . I = h . (f . I) by FUNCT_1:13;
A6: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) = ((2 * d) / (d - c)) + (- ((d + c) / (d - c))) by A3, XCMPLX_1:74
.= ((2 * d) / (d - c)) + ((- (d + c)) / (d - c)) by XCMPLX_1:187
.= ((2 * d) + (- (d + c))) / (d - c) by XCMPLX_1:62
.= 1 by A4, XCMPLX_1:60 ;
hence ((h * f) . I) `2 = 1 by A5, A6, EUCLID:52; ::_thesis: verum
end;
theorem Th56: :: JGRAPH_7:56
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `1 = b holds
((h * f) . I) `1 = 1
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `1 = b holds
((h * f) . I) `1 = 1
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `1 = b holds
((h * f) . I) `1 = 1
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `1 = b holds
((h * f) . I) `1 = 1
let I be Point of I[01]; ::_thesis: ( a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `1 = b implies ((h * f) . I) `1 = 1 )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: (f . I) `1 = b ; ::_thesis: ((h * f) . I) `1 = 1
A4: b - a > 0 by A1, XREAL_1:50;
dom f = the carrier of I[01] by FUNCT_2:def_1;
then A5: (h * f) . I = h . (f . I) by FUNCT_1:13;
A6: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) = ((2 * b) / (b - a)) + (- ((b + a) / (b - a))) by A3, XCMPLX_1:74
.= ((2 * b) / (b - a)) + ((- (b + a)) / (b - a)) by XCMPLX_1:187
.= ((b + b) + (- (b + a))) / (b - a) by XCMPLX_1:62
.= 1 by A4, XCMPLX_1:60 ;
hence ((h * f) . I) `1 = 1 by A5, A6, EUCLID:52; ::_thesis: verum
end;
theorem Th57: :: JGRAPH_7:57
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = c holds
((h * f) . I) `2 = - 1
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = c holds
((h * f) . I) `2 = - 1
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = c holds
((h * f) . I) `2 = - 1
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = c holds
((h * f) . I) `2 = - 1
let I be Point of I[01]; ::_thesis: ( c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & (f . I) `2 = c implies ((h * f) . I) `2 = - 1 )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: c < d and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: (f . I) `2 = c ; ::_thesis: ((h * f) . I) `2 = - 1
A4: d - c > 0 by A1, XREAL_1:50;
dom f = the carrier of I[01] by FUNCT_2:def_1;
then A5: (h * f) . I = h . (f . I) by FUNCT_1:13;
A6: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) = ((2 * c) / (d - c)) + (- ((d + c) / (d - c))) by A3, XCMPLX_1:74
.= ((2 * c) / (d - c)) + ((- (d + c)) / (d - c)) by XCMPLX_1:187
.= ((c + c) + (- (d + c))) / (d - c) by XCMPLX_1:62
.= (- (d - c)) / (d - c)
.= - 1 by A4, XCMPLX_1:197 ;
hence ((h * f) . I) `2 = - 1 by A5, A6, EUCLID:52; ::_thesis: verum
end;
theorem Th58: :: JGRAPH_7:58
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d implies ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: c < d and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: c <= (f . O) `2 and
A4: (f . O) `2 < (f . I) `2 and
A5: (f . I) `2 <= d ; ::_thesis: ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
A6: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
A7: d - c > 0 by A1, XREAL_1:50;
then A8: 2 / (d - c) > 0 by XREAL_1:139;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A7, XCMPLX_1:113
.= (((c + c) / (d - c)) / 2) * (d - c) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A7, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A3, A8, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A8, XCMPLX_1:87;
then A9: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A10: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A11: (h * f) . O = h . (f . O) by FUNCT_1:13;
A12: (h * f) . I = h . (f . I) by A10, FUNCT_1:13;
(1 - (- ((d + c) / (d - c)))) / (2 / (d - c)) = (1 + ((d + c) / (d - c))) / (2 / (d - c))
.= (((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A7, XCMPLX_1:113
.= (((d + d) / (d - c)) / 2) * (d - c) by XCMPLX_1:82
.= ((d - c) * ((d + d) / (d - c))) / 2
.= (d + d) / 2 by A7, XCMPLX_1:87
.= d ;
then (2 / (d - c)) * ((1 - (- ((d + c) / (d - c)))) / (2 / (d - c))) >= (2 / (d - c)) * ((f . I) `2) by A5, A8, XREAL_1:64;
then 1 - (- ((d + c) / (d - c))) >= (2 / (d - c)) * ((f . I) `2) by A8, XCMPLX_1:87;
then A13: (1 - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A14: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
(2 / (d - c)) * ((f . O) `2) < (2 / (d - c)) * ((f . I) `2) by A4, A8, XREAL_1:68;
then ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) < ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:8;
then ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) < ((h * f) . I) `2 by A12, A14, EUCLID:52;
hence ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 ) by A11, A12, A6, A14, A9, A13, EUCLID:52; ::_thesis: verum
end;
theorem Th59: :: JGRAPH_7:59
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a <= (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a <= (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a <= (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a <= (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a <= (f . I) `1 & (f . I) `1 <= b implies ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: c < d and
A3: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A4: c <= (f . O) `2 and
A5: (f . O) `2 <= d and
A6: a <= (f . I) `1 and
A7: (f . I) `1 <= b ; ::_thesis: ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
A8: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
A9: d - c > 0 by A2, XREAL_1:50;
then A10: 2 / (d - c) > 0 by XREAL_1:139;
then (2 / (d - c)) * d >= (2 / (d - c)) * ((f . O) `2) by A5, XREAL_1:64;
then A11: ((2 / (d - c)) * d) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A9, XCMPLX_1:113
.= (((c + c) / (d - c)) / 2) * (d - c) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A9, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A4, A10, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A10, XCMPLX_1:87;
then A12: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A13: ((2 / (d - c)) * d) + (- ((d + c) / (d - c))) = ((2 * d) / (d - c)) + (- ((d + c) / (d - c))) by XCMPLX_1:74
.= ((2 * d) / (d - c)) + ((- (d + c)) / (d - c)) by XCMPLX_1:187
.= ((2 * d) + (- (d + c))) / (d - c) by XCMPLX_1:62
.= 1 by A9, XCMPLX_1:60 ;
A14: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
A15: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A16: (h * f) . I = h . (f . I) by FUNCT_1:13;
A17: b - a > 0 by A1, XREAL_1:50;
then A18: 2 / (b - a) > 0 by XREAL_1:139;
then (2 / (b - a)) * b >= (2 / (b - a)) * ((f . I) `1) by A7, XREAL_1:64;
then A19: ((2 / (b - a)) * b) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
(2 / (b - a)) * a <= (2 / (b - a)) * ((f . I) `1) by A6, A18, XREAL_1:64;
then A20: ((2 / (b - a)) * a) + (- ((b + a) / (b - a))) <= ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:7;
A21: ((2 / (b - a)) * b) + (- ((b + a) / (b - a))) = ((2 * b) / (b - a)) + (- ((b + a) / (b - a))) by XCMPLX_1:74
.= ((2 * b) / (b - a)) + ((- (b + a)) / (b - a)) by XCMPLX_1:187
.= ((2 * b) + (- (b + a))) / (b - a) by XCMPLX_1:62
.= 1 by A17, XCMPLX_1:60 ;
A22: ((2 / (b - a)) * a) + (- ((b + a) / (b - a))) = ((2 * a) / (b - a)) + (- ((b + a) / (b - a))) by XCMPLX_1:74
.= ((2 * a) / (b - a)) + ((- (b + a)) / (b - a)) by XCMPLX_1:187
.= ((2 * a) + (- (b + a))) / (b - a) by XCMPLX_1:62
.= (- (b - a)) / (b - a)
.= - 1 by A17, XCMPLX_1:197 ;
(h * f) . O = h . (f . O) by A15, FUNCT_1:13;
hence ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) by A16, A8, A14, A22, A21, A13, A12, A11, A19, A20, EUCLID:52; ::_thesis: verum
end;
theorem Th60: :: JGRAPH_7:60
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d implies ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: c < d and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: c <= (f . O) `2 and
A4: (f . O) `2 <= d and
A5: c <= (f . I) `2 and
A6: (f . I) `2 <= d ; ::_thesis: ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
A7: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
A8: d - c > 0 by A1, XREAL_1:50;
then A9: 2 / (d - c) > 0 by XREAL_1:139;
then (2 / (d - c)) * d >= (2 / (d - c)) * ((f . O) `2) by A4, XREAL_1:64;
then A10: ((2 / (d - c)) * d) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A8, XCMPLX_1:113
.= (d - c) * (((c + c) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A8, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A3, A9, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A9, XCMPLX_1:87;
then A11: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
(2 / (d - c)) * c <= (2 / (d - c)) * ((f . I) `2) by A5, A9, XREAL_1:64;
then A12: ((2 / (d - c)) * c) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:7;
A13: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
A14: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A15: (h * f) . I = h . (f . I) by FUNCT_1:13;
(2 / (d - c)) * d >= (2 / (d - c)) * ((f . I) `2) by A6, A9, XREAL_1:64;
then A16: ((2 / (d - c)) * d) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A17: ((2 / (d - c)) * d) + (- ((d + c) / (d - c))) = ((2 * d) / (d - c)) + (- ((d + c) / (d - c))) by XCMPLX_1:74
.= ((2 * d) / (d - c)) + ((- (d + c)) / (d - c)) by XCMPLX_1:187
.= ((d + d) + (- (d + c))) / (d - c) by XCMPLX_1:62
.= 1 by A8, XCMPLX_1:60 ;
A18: ((2 / (d - c)) * c) + (- ((d + c) / (d - c))) = ((2 * c) / (d - c)) + (- ((d + c) / (d - c))) by XCMPLX_1:74
.= ((2 * c) / (d - c)) + ((- (d + c)) / (d - c)) by XCMPLX_1:187
.= ((c + c) + (- (d + c))) / (d - c) by XCMPLX_1:62
.= (- (d - c)) / (d - c)
.= - 1 by A8, XCMPLX_1:197 ;
(h * f) . O = h . (f . O) by A14, FUNCT_1:13;
hence ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 ) by A15, A7, A13, A18, A17, A11, A10, A16, A12, EUCLID:52; ::_thesis: verum
end;
theorem Th61: :: JGRAPH_7:61
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b implies ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: c < d and
A3: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A4: c <= (f . O) `2 and
A5: (f . O) `2 <= d and
A6: a < (f . I) `1 and
A7: (f . I) `1 <= b ; ::_thesis: ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
A8: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
A9: d - c > 0 by A2, XREAL_1:50;
then A10: 2 / (d - c) > 0 by XREAL_1:139;
then (2 / (d - c)) * d >= (2 / (d - c)) * ((f . O) `2) by A5, XREAL_1:64;
then A11: ((2 / (d - c)) * d) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A9, XCMPLX_1:113
.= (d - c) * (((c + c) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A9, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A4, A10, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A10, XCMPLX_1:87;
then A12: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A13: ((2 / (d - c)) * d) + (- ((d + c) / (d - c))) = ((2 * d) / (d - c)) + (- ((d + c) / (d - c))) by XCMPLX_1:74
.= ((2 * d) / (d - c)) + ((- (d + c)) / (d - c)) by XCMPLX_1:187
.= ((d + d) + (- (d + c))) / (d - c) by XCMPLX_1:62
.= 1 by A9, XCMPLX_1:60 ;
A14: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
A15: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A16: (h * f) . I = h . (f . I) by FUNCT_1:13;
A17: b - a > 0 by A1, XREAL_1:50;
then A18: 2 / (b - a) > 0 by XREAL_1:139;
then (2 / (b - a)) * b >= (2 / (b - a)) * ((f . I) `1) by A7, XREAL_1:64;
then A19: ((2 / (b - a)) * b) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
(2 / (b - a)) * a < (2 / (b - a)) * ((f . I) `1) by A6, A18, XREAL_1:68;
then A20: ((2 / (b - a)) * a) + (- ((b + a) / (b - a))) < ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:8;
A21: ((2 / (b - a)) * b) + (- ((b + a) / (b - a))) = ((2 * b) / (b - a)) + (- ((b + a) / (b - a))) by XCMPLX_1:74
.= ((2 * b) / (b - a)) + ((- (b + a)) / (b - a)) by XCMPLX_1:187
.= ((b + b) + (- (b + a))) / (b - a) by XCMPLX_1:62
.= 1 by A17, XCMPLX_1:60 ;
A22: ((2 / (b - a)) * a) + (- ((b + a) / (b - a))) = ((2 * a) / (b - a)) + (- ((b + a) / (b - a))) by XCMPLX_1:74
.= ((2 * a) / (b - a)) + ((- (b + a)) / (b - a)) by XCMPLX_1:187
.= ((a + a) + (- (b + a))) / (b - a) by XCMPLX_1:62
.= (- (b - a)) / (b - a)
.= - 1 by A17, XCMPLX_1:197 ;
(h * f) . O = h . (f . O) by A15, FUNCT_1:13;
hence ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) by A16, A8, A14, A22, A13, A21, A12, A11, A19, A20, EUCLID:52; ::_thesis: verum
end;
theorem Th62: :: JGRAPH_7:62
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 < (f . I) `1 & (f . I) `1 <= b implies ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: a <= (f . O) `1 and
A4: (f . O) `1 < (f . I) `1 and
A5: (f . I) `1 <= b ; ::_thesis: ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
A6: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
A7: b - a > 0 by A1, XREAL_1:50;
then A8: 2 / (b - a) > 0 by XREAL_1:139;
(1 - (- ((b + a) / (b - a)))) / (2 / (b - a)) = (1 + ((b + a) / (b - a))) / (2 / (b - a))
.= (((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A7, XCMPLX_1:113
.= (b - a) * (((b + b) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((b + b) / (b - a))) / 2
.= (b + b) / 2 by A7, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . I) `1) by A5, A8, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . I) `1) by A8, XCMPLX_1:87;
then A9: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A10: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A11: (h * f) . O = h . (f . O) by FUNCT_1:13;
A12: (h * f) . I = h . (f . I) by A10, FUNCT_1:13;
((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a)) = ((- 1) + ((b + a) / (b - a))) / (2 / (b - a))
.= ((((- 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A7, XCMPLX_1:113
.= (b - a) * (((a + a) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((a + a) / (b - a))) / 2
.= (a + a) / 2 by A7, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A3, A8, XREAL_1:64;
then (- 1) - (- ((b + a) / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A8, XCMPLX_1:87;
then A13: ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) <= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A14: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
(2 / (b - a)) * ((f . O) `1) < (2 / (b - a)) * ((f . I) `1) by A4, A8, XREAL_1:68;
then ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) < ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:8;
then ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) < ((h * f) . I) `1 by A12, A14, EUCLID:52;
hence ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) by A11, A12, A6, A14, A13, A9, EUCLID:52; ::_thesis: verum
end;
theorem Th63: :: JGRAPH_7:63
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d implies ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: c < d and
A3: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A4: a <= (f . O) `1 and
A5: (f . O) `1 <= b and
A6: c <= (f . I) `2 and
A7: (f . I) `2 <= d ; ::_thesis: ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
A8: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
A9: b - a > 0 by A1, XREAL_1:50;
then A10: 2 / (b - a) > 0 by XREAL_1:139;
((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a)) = ((- 1) + ((b + a) / (b - a))) / (2 / (b - a))
.= ((((- 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A9, XCMPLX_1:113
.= (b - a) * (((a + a) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((a + a) / (b - a))) / 2
.= (a + a) / 2 by A9, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A4, A10, XREAL_1:64;
then (- 1) - (- ((b + a) / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A10, XCMPLX_1:87;
then A11: ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) <= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A12: d - c > 0 by A2, XREAL_1:50;
then A13: 2 / (d - c) > 0 by XREAL_1:139;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A12, XCMPLX_1:113
.= (d - c) * (((c + c) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A12, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . I) `2) by A6, A13, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . I) `2) by A13, XCMPLX_1:87;
then A14: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A15: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A16: (h * f) . I = h . (f . I) by FUNCT_1:13;
(1 - (- ((b + a) / (b - a)))) / (2 / (b - a)) = (1 + ((b + a) / (b - a))) / (2 / (b - a))
.= (((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A9, XCMPLX_1:113
.= (b - a) * (((b + b) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((b + b) / (b - a))) / 2
.= (b + b) / 2 by A9, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A5, A10, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A10, XCMPLX_1:87;
then A17: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A18: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
(1 - (- ((d + c) / (d - c)))) / (2 / (d - c)) = (1 + ((d + c) / (d - c))) / (2 / (d - c))
.= (((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A12, XCMPLX_1:113
.= (d - c) * (((d + d) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((d + d) / (d - c))) / 2
.= (d + d) / 2 by A12, XCMPLX_1:87
.= d ;
then (2 / (d - c)) * ((1 - (- ((d + c) / (d - c)))) / (2 / (d - c))) >= (2 / (d - c)) * ((f . I) `2) by A7, A13, XREAL_1:64;
then 1 - (- ((d + c) / (d - c))) >= (2 / (d - c)) * ((f . I) `2) by A13, XCMPLX_1:87;
then A19: (1 - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
(h * f) . O = h . (f . O) by A15, FUNCT_1:13;
hence ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 ) by A16, A8, A18, A11, A17, A19, A14, EUCLID:52; ::_thesis: verum
end;
theorem Th64: :: JGRAPH_7:64
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b implies ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: a <= (f . O) `1 and
A4: (f . O) `1 <= b and
A5: a < (f . I) `1 and
A6: (f . I) `1 <= b ; ::_thesis: ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
A7: b - a > 0 by A1, XREAL_1:50;
then A8: 2 / (b - a) > 0 by XREAL_1:139;
A9: (1 - (- ((b + a) / (b - a)))) / (2 / (b - a)) = (1 + ((b + a) / (b - a))) / (2 / (b - a))
.= (((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A7, XCMPLX_1:113
.= (b - a) * (((b + b) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((b + b) / (b - a))) / 2
.= (b + b) / 2 by A7, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A4, A8, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A8, XCMPLX_1:87;
then A10: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
(2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . I) `1) by A6, A8, A9, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . I) `1) by A8, XCMPLX_1:87;
then A11: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A12: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
A13: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
A14: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A15: (h * f) . I = h . (f . I) by FUNCT_1:13;
A16: ((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a)) = ((- 1) + ((b + a) / (b - a))) / (2 / (b - a))
.= ((((- 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A7, XCMPLX_1:113
.= (b - a) * (((a + a) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((a + a) / (b - a))) / 2
.= (a + a) / 2 by A7, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A3, A8, XREAL_1:64;
then (- 1) - (- ((b + a) / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A8, XCMPLX_1:87;
then A17: ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) <= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
(2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) < (2 / (b - a)) * ((f . I) `1) by A5, A8, A16, XREAL_1:68;
then (- 1) - (- ((b + a) / (b - a))) < (2 / (b - a)) * ((f . I) `1) by A8, XCMPLX_1:87;
then A18: ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) < ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:8;
(h * f) . O = h . (f . O) by A14, FUNCT_1:13;
hence ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) by A15, A13, A12, A17, A10, A11, A18, EUCLID:52; ::_thesis: verum
end;
theorem Th65: :: JGRAPH_7:65
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & d >= (f . O) `2 & (f . O) `2 > (f . I) `2 & (f . I) `2 >= c holds
( 1 >= ((h * f) . O) `2 & ((h * f) . O) `2 > ((h * f) . I) `2 & ((h * f) . I) `2 >= - 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & d >= (f . O) `2 & (f . O) `2 > (f . I) `2 & (f . I) `2 >= c holds
( 1 >= ((h * f) . O) `2 & ((h * f) . O) `2 > ((h * f) . I) `2 & ((h * f) . I) `2 >= - 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & d >= (f . O) `2 & (f . O) `2 > (f . I) `2 & (f . I) `2 >= c holds
( 1 >= ((h * f) . O) `2 & ((h * f) . O) `2 > ((h * f) . I) `2 & ((h * f) . I) `2 >= - 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & d >= (f . O) `2 & (f . O) `2 > (f . I) `2 & (f . I) `2 >= c holds
( 1 >= ((h * f) . O) `2 & ((h * f) . O) `2 > ((h * f) . I) `2 & ((h * f) . I) `2 >= - 1 )
let O, I be Point of I[01]; ::_thesis: ( c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & d >= (f . O) `2 & (f . O) `2 > (f . I) `2 & (f . I) `2 >= c implies ( 1 >= ((h * f) . O) `2 & ((h * f) . O) `2 > ((h * f) . I) `2 & ((h * f) . I) `2 >= - 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: c < d and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: d >= (f . O) `2 and
A4: (f . O) `2 > (f . I) `2 and
A5: (f . I) `2 >= c ; ::_thesis: ( 1 >= ((h * f) . O) `2 & ((h * f) . O) `2 > ((h * f) . I) `2 & ((h * f) . I) `2 >= - 1 )
A6: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
A7: d - c > 0 by A1, XREAL_1:50;
then A8: 2 / (d - c) > 0 by XREAL_1:139;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A7, XCMPLX_1:113
.= (d - c) * (((c + c) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A7, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . I) `2) by A5, A8, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . I) `2) by A8, XCMPLX_1:87;
then A9: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A10: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A11: (h * f) . O = h . (f . O) by FUNCT_1:13;
A12: (h * f) . I = h . (f . I) by A10, FUNCT_1:13;
(1 - (- ((d + c) / (d - c)))) / (2 / (d - c)) = (1 + ((d + c) / (d - c))) / (2 / (d - c))
.= (((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A7, XCMPLX_1:113
.= (d - c) * (((d + d) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((d + d) / (d - c))) / 2
.= (d + d) / 2 by A7, XCMPLX_1:87
.= d ;
then (2 / (d - c)) * ((1 - (- ((d + c) / (d - c)))) / (2 / (d - c))) >= (2 / (d - c)) * ((f . O) `2) by A3, A8, XREAL_1:64;
then 1 - (- ((d + c) / (d - c))) >= (2 / (d - c)) * ((f . O) `2) by A8, XCMPLX_1:87;
then A13: (1 - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A14: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
(2 / (d - c)) * ((f . O) `2) > (2 / (d - c)) * ((f . I) `2) by A4, A8, XREAL_1:68;
then ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) > ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:8;
then ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) > ((h * f) . I) `2 by A12, A14, EUCLID:52;
hence ( 1 >= ((h * f) . O) `2 & ((h * f) . O) `2 > ((h * f) . I) `2 & ((h * f) . I) `2 >= - 1 ) by A11, A12, A6, A14, A13, A9, EUCLID:52; ::_thesis: verum
end;
theorem Th66: :: JGRAPH_7:66
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & c <= (f . O) `2 & (f . O) `2 <= d & a < (f . I) `1 & (f . I) `1 <= b implies ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: c < d and
A3: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A4: c <= (f . O) `2 and
A5: (f . O) `2 <= d and
A6: a < (f . I) `1 and
A7: (f . I) `1 <= b ; ::_thesis: ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
A8: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
A9: b - a > 0 by A1, XREAL_1:50;
then A10: 2 / (b - a) > 0 by XREAL_1:139;
((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a)) = ((- 1) + ((b + a) / (b - a))) / (2 / (b - a))
.= ((((- 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A9, XCMPLX_1:113
.= (b - a) * (((a + a) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((a + a) / (b - a))) / 2
.= (a + a) / 2 by A9, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) < (2 / (b - a)) * ((f . I) `1) by A6, A10, XREAL_1:68;
then (- 1) - (- ((b + a) / (b - a))) < (2 / (b - a)) * ((f . I) `1) by A10, XCMPLX_1:87;
then A11: ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) < ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:8;
A12: d - c > 0 by A2, XREAL_1:50;
then A13: 2 / (d - c) > 0 by XREAL_1:139;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A12, XCMPLX_1:113
.= (d - c) * (((c + c) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A12, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A4, A13, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . O) `2) by A13, XCMPLX_1:87;
then A14: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A15: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A16: (h * f) . I = h . (f . I) by FUNCT_1:13;
(1 - (- ((b + a) / (b - a)))) / (2 / (b - a)) = (1 + ((b + a) / (b - a))) / (2 / (b - a))
.= (((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A9, XCMPLX_1:113
.= (b - a) * (((b + b) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((b + b) / (b - a))) / 2
.= (b + b) / 2 by A9, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . I) `1) by A7, A10, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . I) `1) by A10, XCMPLX_1:87;
then A17: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A18: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def_2;
(1 - (- ((d + c) / (d - c)))) / (2 / (d - c)) = (1 + ((d + c) / (d - c))) / (2 / (d - c))
.= (((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A12, XCMPLX_1:113
.= (d - c) * (((d + d) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((d + d) / (d - c))) / 2
.= (d + d) / 2 by A12, XCMPLX_1:87
.= d ;
then (2 / (d - c)) * ((1 - (- ((d + c) / (d - c)))) / (2 / (d - c))) >= (2 / (d - c)) * ((f . O) `2) by A5, A13, XREAL_1:64;
then 1 - (- ((d + c) / (d - c))) >= (2 / (d - c)) * ((f . O) `2) by A13, XCMPLX_1:87;
then A19: (1 - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
(h * f) . O = h . (f . O) by A15, FUNCT_1:13;
hence ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) by A16, A8, A18, A14, A19, A17, A11, EUCLID:52; ::_thesis: verum
end;
theorem Th67: :: JGRAPH_7:67
for a, b, c, d being real number
for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a < (f . I) `1 & (f . I) `1 < (f . O) `1 & (f . O) `1 <= b holds
( - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 < ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 )
proof
let a, b, c, d be real number ; ::_thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a < (f . I) `1 & (f . I) `1 < (f . O) `1 & (f . O) `1 <= b holds
( - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 < ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a < (f . I) `1 & (f . I) `1 < (f . O) `1 & (f . O) `1 <= b holds
( - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 < ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 )
let f be Function of I[01],(TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a < (f . I) `1 & (f . I) `1 < (f . O) `1 & (f . O) `1 <= b holds
( - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 < ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 )
let O, I be Point of I[01]; ::_thesis: ( a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a < (f . I) `1 & (f . I) `1 < (f . O) `1 & (f . O) `1 <= b implies ( - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 < ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 ) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: a < (f . I) `1 and
A4: (f . I) `1 < (f . O) `1 and
A5: (f . O) `1 <= b ; ::_thesis: ( - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 < ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 )
A6: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
A7: b - a > 0 by A1, XREAL_1:50;
then A8: 2 / (b - a) > 0 by XREAL_1:139;
(1 - (- ((b + a) / (b - a)))) / (2 / (b - a)) = (1 + ((b + a) / (b - a))) / (2 / (b - a))
.= (((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A7, XCMPLX_1:113
.= (b - a) * (((b + b) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((b + b) / (b - a))) / 2
.= (b + b) / 2 by A7, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A5, A8, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A8, XCMPLX_1:87;
then A9: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A10: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A11: (h * f) . O = h . (f . O) by FUNCT_1:13;
A12: (h * f) . I = h . (f . I) by A10, FUNCT_1:13;
((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a)) = ((- 1) + ((b + a) / (b - a))) / (2 / (b - a))
.= ((((- 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A7, XCMPLX_1:113
.= (b - a) * (((a + a) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((a + a) / (b - a))) / 2
.= (a + a) / 2 by A7, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) < (2 / (b - a)) * ((f . I) `1) by A3, A8, XREAL_1:68;
then (- 1) - (- ((b + a) / (b - a))) < (2 / (b - a)) * ((f . I) `1) by A8, XCMPLX_1:87;
then A13: ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) < ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:8;
A14: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def_2;
(2 / (b - a)) * ((f . O) `1) > (2 / (b - a)) * ((f . I) `1) by A4, A8, XREAL_1:68;
then ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) > ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:8;
then ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) > ((h * f) . I) `1 by A12, A14, EUCLID:52;
hence ( - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 < ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 ) by A11, A12, A6, A14, A9, A13, EUCLID:52; ::_thesis: verum
end;
theorem Th68: :: JGRAPH_7:68
for p1, p2, p3, p4 being 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 `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
proof
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 = 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_1:13;
A29: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A15, A27, FUNCT_1:13;
d - c > 0 by A2, XREAL_1:50;
then A30: 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) ;
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;
set x = the 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_1:13;
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, 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_1:13;
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_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A27, A45, A46, FUNCT_1:13;
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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, 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 A38, A49, FUNCT_1:def_3;
A52: g . z2 in rng g by A27, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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_2:5;
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;
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 A38, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A47, A54, A53, A48, FUNCT_1:def_4;
hence rng f meets rng g by A51, A52, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:69
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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: P is_an_arc_of p1,p3 and
A13: Q is_an_arc_of p2,p4 and
A14: P c= closed_inside_of_rectangle (a,b,c,d) and
A15: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A16: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A13, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A12, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A14, A15, A16, Th68; ::_thesis: verum
end;
theorem Th70: :: JGRAPH_7:70
for p1, p2, p3, p4 being 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 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & 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
proof
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 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & 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 ; ::_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 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & 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 `1 = a and
A6: p4 `2 = d and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 < p3 `2 and
A10: p3 `2 <= d and
A11: a <= p4 `1 and
A12: p4 `1 <= b 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) `1 by A1, A2, A6, A11, A12, A16, Th59;
A23: (g2 . I) `2 = 1 by A2, A6, A16, Th55;
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) `1 <= 1 by A1, A2, A6, A11, A12, A16, Th59;
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, Th15, 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
end;
theorem :: JGRAPH_7:71
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a <= p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a <= p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a <= p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a <= p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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 `2 = d and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 < p3 `2 and
A10: p3 `2 <= d and
A11: a <= p4 `1 and
A12: p4 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th70; ::_thesis: verum
end;
theorem Th72: :: JGRAPH_7:72
for p1, p2, p3, p4 being 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 `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
proof
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
end;
theorem :: JGRAPH_7:73
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th72; ::_thesis: verum
end;
theorem Th74: :: JGRAPH_7:74
for p1, p2, p3, p4 being 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 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & 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
proof
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 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & 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 ; ::_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 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & 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 `1 = a and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 < p3 `2 and
A10: p3 `2 <= d and
A11: a < p4 `1 and
A12: p4 `1 <= b 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) `1 by A1, A2, A6, A11, A12, A16, Th61;
A23: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
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) `1 <= 1 by A1, A2, A6, A11, A12, A16, Th61;
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, Th17, 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
end;
theorem :: JGRAPH_7:75
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = a & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 < p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 < p3 `2 and
A10: p3 `2 <= d and
A11: a < p4 `1 and
A12: p4 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th74; ::_thesis: verum
end;
theorem Th76: :: JGRAPH_7:76
for p1, p2, p3, p4 being 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 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < 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
proof
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 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < 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 ; ::_thesis: 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 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < 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 = d 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 < p4 `1 and
A12: p4 `1 <= b 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 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) ;
A21: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
c < p2 `2 by A7, A8, XXREAL_0:2;
then A22: (g2 . O) `2 <= 1 by A1, A2, A4, A9, A15, Th59;
A23: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A24: (g2 . I) `2 = 1 by A2, A6, A16, Th55;
d - c > 0 by A2, XREAL_1:50;
then A25: 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) ;
A26: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
p3 `1 < b by A11, A12, XXREAL_0:2;
then A27: - 1 <= (f2 . I) `1 by A1, A2, A5, A10, A14, Th59;
A28: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A29: (f2 . I) `2 = 1 by A2, A5, A14, Th55;
p1 `2 <= d by A8, A9, XXREAL_0:2;
then A30: - 1 <= (f2 . O) `2 by A1, A2, A3, A7, A13, Th59;
A31: (f2 . O) `1 = - 1 by A1, A3, A13, Th54;
set x = the Element of (rng f2) /\ (rng g2);
A32: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A33: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A14, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A34: 2 / (b - a) > 0 by XREAL_1:139;
then A35: ((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, A25, Th51;
a < p4 `1 by A10, A11, XXREAL_0:2;
then A36: (g2 . I) `1 <= 1 by A1, A2, A6, A12, A16, Th59;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A34, A25, Th51;
then A37: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A38: (g2 . O) `1 = - 1 by A1, A4, A15, Th54;
A39: dom g = the carrier of I[01] by FUNCT_2:def_1;
then A40: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, FUNCT_1:13;
A41: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 by A11, A34, A25, Th50;
A42: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A39, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A13, A32, FUNCT_1:13;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A35, A41, A40, A33, A42, A31, A29, A30, A27, A38, A24, A22, A36, Th18;
then rng f2 meets rng g2 by A26, A28, A21, A23, 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 A39, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A39, 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 A32, 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 A32, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A37, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:77
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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 = d 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 < p4 `1 and
A12: p4 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th76; ::_thesis: verum
end;
theorem Th78: :: JGRAPH_7:78
for p1, p2, p3, p4 being 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 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & 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
proof
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 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & 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 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & 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 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & 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 `2 = d and
A6: p4 `1 = b 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: c <= p4 `2 and
A13: p4 `2 <= d 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) ; ::_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) ;
A22: - 1 <= (f2 . I) `1 by A1, A2, A5, A10, A11, A15, Th59;
A23: (f2 . I) `1 <= 1 by A1, A2, A5, A10, A11, A15, Th59;
A24: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A18, Th53;
A25: (f2 . I) `2 = 1 by A2, A5, A15, Th55;
A26: (f2 . O) `1 = - 1 by A1, A3, A14, Th54;
A27: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
d - c > 0 by A2, XREAL_1:50;
then A28: 2 / (d - c) > 0 by XREAL_1:139;
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) ;
A29: - 1 <= (g2 . I) `2 by A2, A12, A13, A17, Th60;
A30: (g2 . O) `1 = - 1 by A1, A4, A16, Th54;
A31: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A19, Th53;
c < p2 `2 by A7, A8, XXREAL_0:2;
then A32: (g2 . O) `2 <= 1 by A2, A9, A16, Th60;
A33: (g2 . I) `2 <= 1 by A2, A12, A13, A17, Th60;
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))))) . p2 = g2 . O by A16, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A36: 2 / (b - a) > 0 by XREAL_1:139;
then A37: ((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;
p1 `2 <= d by A8, A9, XXREAL_0:2;
then A38: - 1 <= (f2 . O) `2 by A1, A2, A3, A7, A14, Th59;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A36, A28, Th51;
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: (g2 . I) `1 = 1 by A1, A6, A17, Th56;
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 A37, A35, A24, A26, A25, A38, A22, A23, A27, A31, A30, A41, A32, A29, A33, A40, Th19, 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 A34, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A34, 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 A42, 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 A42, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A39, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:79
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a <= p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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 `1 = b 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: c <= p4 `2 and
A13: p4 `2 <= d and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th78; ::_thesis: verum
end;
theorem Th80: :: JGRAPH_7:80
for p1, p2, p3, p4 being 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
proof
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 `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 ; ::_thesis: 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); ::_thesis: ( 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) ; ::_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) ;
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; ::_thesis: verum
end;
theorem :: JGRAPH_7:81
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( 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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th80; ::_thesis: verum
end;
theorem Th82: :: JGRAPH_7:82
for p1, p2, p3, p4 being 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 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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
proof
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 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: c <= p4 `2 and
A11: p4 `2 < p3 `2 and
A12: p3 `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 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) ;
A21: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
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) ;
A22: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
A23: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A24: (f2 . I) `1 = 1 by A1, A5, A14, Th56;
c < p2 `2 by A7, A8, XXREAL_0:2;
then A25: (g2 . O) `2 <= 1 by A2, A9, A15, Th60;
p3 `2 > c by A10, A11, XXREAL_0:2;
then A26: (f2 . I) `2 <= 1 by A2, A12, A14, Th60;
d - c > 0 by A2, XREAL_1:50;
then A27: 2 / (d - c) > 0 by XREAL_1:139;
A28: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A29: (g2 . I) `1 = 1 by A1, A6, A16, Th56;
A30: (g2 . O) `1 = - 1 by A1, A4, A15, Th54;
A31: dom g = the carrier of I[01] by FUNCT_2:def_1;
then A32: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, FUNCT_1:13;
p1 `2 <= d by A8, A9, XXREAL_0:2;
then A33: - 1 <= (f2 . O) `2 by A2, A7, A13, Th60;
A34: (f2 . O) `1 = - 1 by A1, A3, A13, Th54;
set x = the Element of (rng f2) /\ (rng g2);
A35: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A36: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A14, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A37: 2 / (b - a) > 0 by XREAL_1:139;
then A38: ((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, A27, Th51;
d >= p4 `2 by A11, A12, XXREAL_0:2;
then A39: - 1 <= (g2 . I) `2 by A2, A10, A16, Th60;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A37, A27, Th51;
then A40: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A41: ((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 A11, A37, A27, Th51;
A42: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A31, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A13, A35, FUNCT_1:13;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A38, A41, A32, A36, A42, A34, A24, A33, A26, A30, A29, A25, A39, Th21;
then rng f2 meets rng g2 by A22, A23, A21, A28, 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 A31, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A31, 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 A35, 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 A35, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A40, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:83
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = a and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: c <= p4 `2 and
A11: p4 `2 < p3 `2 and
A12: p3 `2 <= d and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th82; ::_thesis: verum
end;
theorem Th84: :: JGRAPH_7:84
for p1, p2, p3, p4 being 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 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & 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
proof
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 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & 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 ; ::_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 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & 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 `1 = b and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: c <= p3 `2 and
A11: p3 `2 <= d 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) ; ::_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) ;
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) `2 by A2, A10, A11, A15, Th60;
A30: (f2 . O) `1 = - 1 by A1, A3, A14, Th54;
A31: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A18, Th53;
p1 `2 <= d by A8, A9, XXREAL_0:2;
then A32: - 1 <= (f2 . O) `2 by A2, A7, A14, Th60;
A33: (f2 . I) `2 <= 1 by A2, A10, A11, A15, Th60;
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))))) . p2 = g2 . O by A16, FUNCT_1:13;
A36: (g . I) `2 = c by A6, A17;
c < p2 `2 by A7, A8, XXREAL_0:2;
then A37: (g2 . O) `2 <= 1 by A1, A2, A9, A12, A13, A16, A17, A36, Th61;
b - a > 0 by A1, XREAL_1:50;
then A38: 2 / (b - a) > 0 by XREAL_1:139;
then A39: ((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;
A40: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A41: (f2 . I) `1 = 1 by A1, A5, A15, Th56;
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 A39, A35, A31, A30, A41, A32, A29, A33, A40, A24, A26, A25, A37, A22, A23, A27, Th22, 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 A34, A44, A45, FUNCT_1:13;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A38, 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 A34, 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; ::_thesis: verum
end;
theorem :: JGRAPH_7:85
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = a and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: c <= p3 `2 and
A11: p3 `2 <= d and
A12: a < p4 `1 and
A13: p4 `1 <= b and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th84; ::_thesis: verum
end;
theorem Th86: :: JGRAPH_7:86
for p1, p2, p3, p4 being 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 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < 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
proof
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 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < 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 = a and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: a < p4 `1 and
A11: p4 `1 < p3 `1 and
A12: p3 `1 <= b 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 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) ;
A21: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
A22: b >= p4 `1 by A11, A12, XXREAL_0:2;
then A23: - 1 < (g2 . I) `1 by A1, A2, A6, A10, A16, Th61;
A24: (g . I) `2 = c by A6, A16;
c < p2 `2 by A7, A8, XXREAL_0:2;
then A25: (g2 . O) `2 <= 1 by A1, A2, A9, A10, A15, A16, A22, A24, Th61;
d - c > 0 by A2, XREAL_1:50;
then A26: 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) ;
A27: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
A28: p3 `1 > a by A10, A11, XXREAL_0:2;
then A29: (f2 . I) `1 <= 1 by A1, A2, A5, A12, A14, Th61;
A30: (f . I) `2 = c by A5, A14;
p1 `2 <= d by A8, A9, XXREAL_0:2;
then A31: - 1 <= (f2 . O) `2 by A1, A2, A7, A12, A13, A14, A28, A30, Th61;
A32: (f2 . O) `1 = - 1 by A1, A3, A13, Th54;
set x = the Element of (rng f2) /\ (rng g2);
A33: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A34: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A14, FUNCT_1:13;
A35: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A36: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
A37: (g2 . O) `1 = - 1 by A1, A4, A15, Th54;
A38: dom g = 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))))) . p2 = g2 . O by A15, FUNCT_1:13;
A40: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A41: (f2 . I) `2 = - 1 by A2, A5, A14, Th57;
b - a > 0 by A1, XREAL_1:50;
then A42: 2 / (b - a) > 0 by XREAL_1:139;
then A43: ((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, A26, Th51;
A44: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 by A11, A42, A26, Th50;
A45: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A38, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A13, A33, FUNCT_1:13;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A43, A44, A39, A34, A45, A32, A41, A31, A29, A37, A36, A25, A23, Th23;
then rng f2 meets rng g2 by A27, A40, A21, A35, 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 A38, A47, A48, FUNCT_1:13;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A42, A26, Th51;
then A50: 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 A46, XBOOLE_0:def_4;
then consider z1 being set such that
A51: z1 in dom f2 and
A52: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def_3;
A53: f . z1 in rng f by A33, A51, FUNCT_1:def_3;
A54: g . z2 in rng g by A38, A47, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A47, FUNCT_2:5;
then A55: 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 A51, FUNCT_2:5;
then A56: 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 A33, A51, A52, FUNCT_1:13;
then f . z1 = g . z2 by A49, A56, A55, A50, FUNCT_1:def_4;
hence rng f meets rng g by A53, A54, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:87
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = a & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = a and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 < p2 `2 and
A9: p2 `2 <= d and
A10: a < p4 `1 and
A11: p4 `1 < p3 `1 and
A12: p3 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th86; ::_thesis: verum
end;
theorem Th88: :: JGRAPH_7:88
for p1, p2, p3, p4 being 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 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < 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
proof
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 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < 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 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = d and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < p3 `1 and
A11: p3 `1 < p4 `1 and
A12: p4 `1 <= b 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 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) ;
A21: p2 `1 < p4 `1 by A10, A11, XXREAL_0:2;
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) ;
A22: - 1 <= (f2 . O) `2 by A1, A2, A3, A7, A8, A13, Th59;
A23: (f2 . O) `1 = - 1 by A1, A3, A13, Th54;
A24: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
A25: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A26: (f2 . I) `2 = 1 by A2, A5, A14, Th55;
A27: (f2 . O) `2 <= 1 by A1, A2, A3, A7, A8, A13, Th59;
set x = the Element of (rng f2) /\ (rng g2);
A28: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A29: (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 A30: 2 / (d - c) > 0 by XREAL_1:139;
b - a > 0 by A1, XREAL_1:50;
then A31: 2 / (b - a) > 0 by XREAL_1:139;
then A32: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A10, A30, Th50;
(g . O) `2 = d by A4, A15;
then A33: (g2 . I) `1 <= 1 by A1, A9, A12, A15, A16, A21, Th62;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A31, A30, Th50;
then A34: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A35: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A36: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 by A11, A31, A30, Th50;
A37: (g2 . I) `2 = 1 by A2, A6, A16, Th55;
A38: (g2 . O) `2 = 1 by A2, A4, A15, Th55;
(g . I) `2 = d by A6, A16;
then A39: - 1 <= (g2 . O) `1 by A1, A9, A12, A15, A16, A21, Th62;
A40: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
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))))) . p4 = g2 . I by A16, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, A41, FUNCT_1:13;
then rng f2 meets rng g2 by A32, A36, A29, A42, A24, A23, A26, A22, A27, A25, A40, A38, A37, A39, A33, A35, Th24, 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 A28, 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 A28, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A34, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:89
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = d & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = d and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < p3 `1 and
A11: p3 `1 < p4 `1 and
A12: p4 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th88; ::_thesis: verum
end;
theorem Th90: :: JGRAPH_7:90
for p1, p2, p3, p4 being 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 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & 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
proof
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 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & 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 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & 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 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & 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 `2 = d and
A5: p3 `2 = d and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < p3 `1 and
A11: p3 `1 <= b and
A12: c <= p4 `2 and
A13: p4 `2 <= d 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) ; ::_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) ;
A22: - 1 <= (g2 . I) `2 by A1, A2, A6, A12, A13, A17, Th63;
A23: (g2 . I) `2 <= 1 by A1, A2, A6, A12, A13, A17, Th63;
A24: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A19, Th53;
A25: (g2 . I) `1 = 1 by A1, A6, A17, Th56;
A26: (g2 . O) `2 = 1 by A2, A4, A16, Th55;
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 . O) `2 by A1, A2, A3, A7, A8, A14, 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;
a <= p3 `1 by A9, A10, XXREAL_0:2;
then A32: (f2 . I) `1 <= 1 by A1, A2, A5, A11, A15, 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 . O) `2 <= 1 by A1, A2, A3, A7, A8, A14, Th59;
set x = the Element of (rng f2) /\ (rng g2);
A36: dom f = 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))))) . p3 = f2 . I by A15, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A38: 2 / (b - a) > 0 by XREAL_1:139;
then A39: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A10, A28, Th50;
p2 `1 < p4 `1 by A6, A10, A11, XXREAL_0:2;
then A40: - 1 <= (g2 . O) `1 by A1, A2, A4, A6, A9, A16, Th63;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A38, A28, Th50;
then A41: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A42: dom g = 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))))) . p2 = g2 . O by A16, FUNCT_1:13;
then rng f2 meets rng g2 by A39, A37, A34, A33, A31, A29, A35, A32, A30, A24, A26, A25, A40, A22, A23, A27, Th25, 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 A42, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A42, 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 A36, 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 A36, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A41, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:91
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < p3 `1 and
A11: p3 `1 <= b and
A12: c <= p4 `2 and
A13: p4 `2 <= d and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th90; ::_thesis: verum
end;
theorem Th92: :: JGRAPH_7:92
for p1, p2, p3, p4 being 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 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < 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
proof
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 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < 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 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < 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) ; ::_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) ;
A22: - 1 <= (f2 . O) `2 by A1, A2, A3, A7, A8, A14, Th59;
A23: (f2 . O) `2 <= 1 by A1, A2, A3, A7, A8, A14, Th59;
A24: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A18, Th53;
A25: (f2 . I) `2 = 1 by A2, A5, A15, Th55;
A26: (f2 . O) `1 = - 1 by A1, A3, A14, Th54;
A27: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
d - c > 0 by A2, XREAL_1:50;
then A28: 2 / (d - c) > 0 by XREAL_1:139;
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) ;
A29: - 1 < (g2 . I) `1 by A1, A12, A13, A17, Th64;
A30: (g2 . O) `2 = 1 by A2, A4, A16, Th55;
A31: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A19, Th53;
A32: (g . I) `2 = c by A6, A17;
p2 `1 < b by A10, A11, XXREAL_0:2;
then A33: - 1 <= (g2 . O) `1 by A1, A9, A12, A13, A16, A17, A32, Th64;
A34: (g2 . I) `1 <= 1 by A1, A12, A13, A17, Th64;
set x = the Element of (rng f2) /\ (rng g2);
A35: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A36: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A15, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A37: 2 / (b - a) > 0 by XREAL_1:139;
then A38: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A10, A28, Th50;
a <= p3 `1 by A9, A10, XXREAL_0:2;
then A39: (f2 . I) `1 <= 1 by A1, A2, A5, A11, A15, Th59;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A37, A28, Th50;
then A40: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A41: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A21, Th52;
A42: (g2 . I) `2 = - 1 by A2, A6, A17, Th57;
A43: dom g = 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))))) . p2 = g2 . O by A16, FUNCT_1:13;
then rng f2 meets rng g2 by A38, A36, A24, A26, A25, A22, A23, A39, A27, A31, A30, A42, A33, A29, A34, A41, Th26, JGRAPH_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A43, A45, A46, FUNCT_1:13;
A48: g . z2 in rng g by A43, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, XBOOLE_0:def_4;
then consider z1 being set such that
A50: z1 in dom f2 and
A51: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def_3;
A52: f . z1 in rng f by A35, A50, FUNCT_1:def_3;
f . z1 in the carrier of (TOP-REAL 2) by A50, 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 A35, A50, A51, FUNCT_1:13;
then f . z1 = g . z2 by A47, A53, A49, A40, FUNCT_1:def_4;
hence rng f meets rng g by A52, A48, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:93
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `2 = d & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 < p3 `1 and
A11: p3 `1 <= b and
A12: a < p4 `1 and
A13: p4 `1 <= b and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th92; ::_thesis: verum
end;
theorem Th94: :: JGRAPH_7:94
for p1, p2, p3, p4 being 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 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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
proof
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 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 `2 = d and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 <= b and
A11: c <= p4 `2 and
A12: p4 `2 < p3 `2 and
A13: p3 `2 <= d 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) ; ::_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) ;
A22: - 1 <= (g2 . O) `1 by A1, A2, A4, A9, A10, A16, Th63;
A23: (g2 . O) `1 <= 1 by A1, A2, A4, A9, A10, A16, Th63;
A24: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A19, Th53;
A25: (g2 . I) `1 = 1 by A1, A6, A17, Th56;
A26: (g2 . O) `2 = 1 by A2, A4, A16, Th55;
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 . O) `2 by A2, A7, A8, A14, Th60;
A30: (f2 . O) `1 = - 1 by A1, A3, A14, Th54;
A31: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A18, Th53;
p3 `2 > c by A11, A12, XXREAL_0:2;
then A32: (f2 . I) `2 <= 1 by A2, A13, A15, Th60;
A33: (f2 . O) `2 <= 1 by A2, A7, A8, A14, Th60;
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;
b - a > 0 by A1, XREAL_1:50;
then A36: 2 / (b - a) > 0 by XREAL_1:139;
then A37: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `2 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `2 by A12, A28, Th51;
p4 `2 < d by A12, A13, XXREAL_0:2;
then A38: - 1 <= (g2 . I) `2 by A1, A2, A6, A11, A17, Th63;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A36, A28, Th51;
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 f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A41: (f2 . I) `1 = 1 by A1, A5, A15, Th56;
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))))) . p3 = f2 . I by A15, FUNCT_1:13;
then rng f2 meets rng g2 by A37, A35, A31, A30, A41, A29, A33, A32, A40, A24, A26, A25, A22, A23, A38, A27, Th27, 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 A34, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A34, 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 A42, 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 A42, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A39, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:95
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a <= p2 `1 and
A10: p2 `1 <= b and
A11: c <= p4 `2 and
A12: p4 `2 < p3 `2 and
A13: p3 `2 <= d and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th94; ::_thesis: verum
end;
theorem Th96: :: JGRAPH_7:96
for p1, p2, p3, p4 being 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 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & 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
proof
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 `2 = d & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & 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 `2 = d and
A5: p3 `1 = b 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: c <= p3 `2 and
A12: p3 `2 <= d and
A13: a < p4 `1 and
A14: p4 `1 <= b and
A15: f . 0 = p1 and
A16: f . 1 = p3 and
A17: g . 0 = p2 and
A18: g . 1 = p4 and
A19: ( f is continuous & f is one-to-one ) and
A20: ( g is continuous & g is one-to-one ) and
A21: rng f c= closed_inside_of_rectangle (a,b,c,d) and
A22: 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) ;
A23: - 1 < (g2 . I) `1 by A1, A13, A14, A18, Th64;
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) ;
A24: - 1 <= (f2 . O) `2 by A2, A7, A8, A15, Th60;
A25: (f2 . O) `2 <= 1 by A2, A7, A8, A15, Th60;
A26: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A19, Th53;
A27: - 1 <= (f2 . I) `2 by A2, A11, A12, A16, Th60;
A28: (f2 . O) `1 = - 1 by A1, A3, A15, Th54;
A29: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A21, Th52;
set x = the Element of (rng f2) /\ (rng g2);
d - c > 0 by A2, XREAL_1:50;
then A30: 2 / (d - c) > 0 by XREAL_1:139;
A31: (g . I) `2 = c by A6, A18;
then A32: (g2 . O) `1 <= 1 by A1, A9, A10, A13, A14, A17, A18, Th64;
A33: (g2 . I) `1 <= 1 by A1, A13, A14, A18, Th64;
A34: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A20, Th53;
A35: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A22, Th52;
A36: (g2 . I) `2 = - 1 by A2, A6, A18, Th57;
A37: (g2 . O) `2 = 1 by A2, A4, A17, Th55;
A38: (f2 . I) `2 <= 1 by A2, A11, A12, A16, Th60;
A39: (f2 . I) `1 = 1 by A1, A5, A16, Th56;
- 1 <= (g2 . O) `1 by A1, A9, A10, A13, A14, A17, A18, A31, Th64;
then rng f2 meets rng g2 by A26, A28, A39, A24, A25, A27, A38, A29, A34, A37, A36, A32, A23, A33, A35, Th28, JGRAPH_6:79;
then A40: (rng f2) /\ (rng g2) <> {} by XBOOLE_0:def_7;
then the Element of (rng f2) /\ (rng g2) in rng f2 by XBOOLE_0:def_4;
then consider z1 being set such that
A41: z1 in dom f2 and
A42: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def_3;
A43: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A44: f . z1 in rng f by A41, FUNCT_1:def_3;
b - a > 0 by A1, XREAL_1:50;
then 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 A30, Th51;
then A45: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
f . z1 in the carrier of (TOP-REAL 2) by A41, FUNCT_2:5;
then A46: 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) in rng g2 by A40, 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: dom g = the carrier of I[01] by FUNCT_2:def_1;
then A50: g . z2 in rng g by A47, FUNCT_1:def_3;
A51: 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 A49, A47, A48, FUNCT_1:13;
g . z2 in the carrier of (TOP-REAL 2) by A47, 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;
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 A43, A41, A42, FUNCT_1:13;
then f . z1 = g . z2 by A51, A46, A52, A45, FUNCT_1:def_4;
hence rng f meets rng g by A44, A50, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:97
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = d & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a <= p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = d and
A5: p3 `1 = b 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: c <= p3 `2 and
A12: p3 `2 <= d and
A13: a < p4 `1 and
A14: p4 `1 <= b and
A15: P is_an_arc_of p1,p3 and
A16: Q is_an_arc_of p2,p4 and
A17: P c= closed_inside_of_rectangle (a,b,c,d) and
A18: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A19: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A16, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A15, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A17, A18, A19, Th96; ::_thesis: verum
end;
theorem Th98: :: JGRAPH_7:98
for p1, p2, p3, p4 being 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
proof
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 `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 ; ::_thesis: 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); ::_thesis: ( 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) ; ::_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) ;
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; ::_thesis: verum
end;
theorem :: JGRAPH_7:99
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( 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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th98; ::_thesis: verum
end;
theorem Th100: :: JGRAPH_7:100
for p1, p2, p3, p4 being 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 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 < p2 `2 & p2 `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
proof
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 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 < p2 `2 & p2 `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 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 < p2 `2 & p2 `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 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 < p2 `2 & p2 `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 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: c <= p4 `2 and
A10: p4 `2 < p3 `2 and
A11: p3 `2 < p2 `2 and
A12: p2 `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 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) ;
A21: p2 `2 > p4 `2 by A10, A11, XXREAL_0:2;
(g . O) `1 = b by A4, A15;
then A22: (g2 . I) `2 >= - 1 by A2, A9, A12, A15, A16, A21, Th65;
d - c > 0 by A2, XREAL_1:50;
then A23: 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) ;
A24: (f2 . O) `2 <= 1 by A2, A7, A8, A13, Th60;
A25: (f2 . O) `1 = - 1 by A1, A3, A13, Th54;
A26: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
A27: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A28: (f2 . I) `1 = 1 by A1, A5, A14, Th56;
set x = the Element of (rng f2) /\ (rng g2);
A29: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A30: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A14, FUNCT_1:13;
(g . I) `1 = b by A6, A16;
then A31: 1 >= (g2 . O) `2 by A2, A9, A12, A15, A16, A21, Th65;
A32: (g2 . O) `1 = 1 by A1, A4, A15, Th56;
A33: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
A34: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A35: (g2 . I) `1 = 1 by A1, A6, A16, Th56;
b - a > 0 by A1, XREAL_1:50;
then A36: 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 A23, Th50;
then A37: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A38: dom g = 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))))) . p2 = g2 . O by A15, FUNCT_1:13;
then A39: (g2 . O) `2 > (f2 . I) `2 by A11, A36, A23, A30, Th51;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A38, FUNCT_1:13;
then A40: (g2 . I) `2 < (f2 . I) `2 by A10, A36, A23, A30, Th51;
- 1 <= (f2 . O) `2 by A2, A7, A8, A13, Th60;
then rng f2 meets rng g2 by A26, A25, A28, A24, A27, A33, A32, A35, A31, A22, A34, A39, A40, Th30, JGRAPH_6:79;
then A41: (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
A42: z2 in dom g2 and
A43: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A44: 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 A38, A42, A43, FUNCT_1:13;
A45: g . z2 in rng g by A38, A42, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A42, FUNCT_2:5;
then A46: 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 A41, XBOOLE_0:def_4;
then consider z1 being set such that
A47: z1 in dom f2 and
A48: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def_3;
A49: f . z1 in rng f by A29, A47, FUNCT_1:def_3;
f . z1 in the carrier of (TOP-REAL 2) by A47, FUNCT_2:5;
then A50: 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 A29, A47, A48, FUNCT_1:13;
then f . z1 = g . z2 by A44, A50, A46, A37, FUNCT_1:def_4;
hence rng f meets rng g by A49, A45, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:101
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `1 = b & c <= p1 `2 & p1 `2 <= d & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: c <= p4 `2 and
A10: p4 `2 < p3 `2 and
A11: p3 `2 < p2 `2 and
A12: p2 `2 <= d and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th100; ::_thesis: verum
end;
theorem Th102: :: JGRAPH_7:102
for p1, p2, p3, p4 being 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 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & 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
proof
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 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & 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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & 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 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: c <= p3 `2 and
A10: p3 `2 < p2 `2 and
A11: p2 `2 <= d 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) ; ::_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) ;
A22: - 1 <= (f2 . O) `2 by A2, A7, A8, A14, Th60;
A23: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
d - c > 0 by A2, XREAL_1:50;
then A24: 2 / (d - c) > 0 by XREAL_1:139;
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) ;
A25: (g2 . I) `1 > - 1 by A1, A2, A6, A12, A13, A17, Th66;
A26: (g2 . O) `1 = 1 by A1, A4, A16, Th56;
A27: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A19, Th53;
p2 `2 > c by A9, A10, XXREAL_0:2;
then A28: 1 >= (g2 . O) `2 by A1, A2, A4, A11, A16, Th66;
set x = the Element of (rng f2) /\ (rng g2);
A29: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A30: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A15, FUNCT_1:13;
A31: (f2 . O) `2 <= 1 by A2, A7, A8, A14, Th60;
A32: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A18, Th53;
b - a > 0 by A1, XREAL_1:50;
then A33: 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 A24, Th50;
then A34: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A35: dom g = 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))))) . p2 = g2 . O by A16, FUNCT_1:13;
then A36: (g2 . O) `2 > (f2 . I) `2 by A10, A33, A24, A30, Th51;
p3 `2 < d by A10, A11, XXREAL_0:2;
then A37: - 1 <= (f2 . I) `2 by A2, A9, A15, Th60;
A38: (f2 . I) `1 = 1 by A1, A5, A15, Th56;
A39: (f2 . O) `1 = - 1 by A1, A3, A14, Th54;
A40: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A21, Th52;
A41: (g2 . I) `2 = - 1 by A2, A6, A17, Th57;
1 >= (g2 . I) `1 by A1, A2, A6, A12, A13, A17, Th66;
then rng f2 meets rng g2 by A32, A39, A38, A22, A31, A37, A23, A27, A26, A41, A28, A25, A40, A36, Th31, JGRAPH_6:79;
then A42: (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
A43: z2 in dom g2 and
A44: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A45: 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 A35, A43, A44, FUNCT_1:13;
A46: g . z2 in rng g by A35, A43, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A43, FUNCT_2:5;
then A47: 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 A42, 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 A29, A48, FUNCT_1:def_3;
f . z1 in the carrier of (TOP-REAL 2) by A48, FUNCT_2:5;
then A51: 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 A29, A48, A49, FUNCT_1:13;
then f . z1 = g . z2 by A45, A51, A47, A34, FUNCT_1:def_4;
hence rng f meets rng g by A50, A46, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:103
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `1 = b & p3 `1 = b & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & c <= p3 `2 & p3 `2 < p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: c <= p3 `2 and
A10: p3 `2 < p2 `2 and
A11: p2 `2 <= d and
A12: a < p4 `1 and
A13: p4 `1 <= b and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th102; ::_thesis: verum
end;
theorem Th104: :: JGRAPH_7:104
for p1, p2, p3, p4 being 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
proof
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 = 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 ; ::_thesis: 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); ::_thesis: ( 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) ; ::_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) ;
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_1:50;
then A24: 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) ;
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;
set x = the 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_1:13;
b - a > 0 by A1, XREAL_1:50;
then A32: 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 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_1:13;
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_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A30, A45, A46, FUNCT_1:13;
A48: g . z2 in rng g by A30, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, XBOOLE_0:def_4;
then consider z1 being set such that
A50: z1 in dom f2 and
A51: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def_3;
A52: f . z1 in rng f by A34, A50, FUNCT_1:def_3;
f . z1 in the carrier of (TOP-REAL 2) by A50, 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 A34, A50, A51, FUNCT_1:13;
then f . z1 = g . z2 by A47, A53, A49, A33, FUNCT_1:def_4;
hence rng f meets rng g by A52, A48, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:105
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( 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 & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
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: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th104; ::_thesis: verum
end;
theorem Th106: :: JGRAPH_7:106
for p1, p2, p3, p4 being 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 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `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
proof
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 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `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); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `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 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a < p4 `1 and
A10: p4 `1 < p3 `1 and
A11: p3 `1 < p2 `1 and
A12: p2 `1 <= b 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 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) ;
A21: p2 `1 > p4 `1 by A10, A11, XXREAL_0:2;
(g . O) `2 = c by A4, A15;
then A22: (g2 . I) `1 > - 1 by A1, A9, A12, A15, A16, A21, Th67;
d - c > 0 by A2, XREAL_1:50;
then A23: 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) ;
A24: b > p3 `1 by A11, A12, XXREAL_0:2;
(g . I) `2 = c by A6, A16;
then A25: 1 >= (g2 . O) `1 by A1, A9, A12, A15, A16, A21, Th67;
A26: (g2 . O) `2 = - 1 by A2, A4, A15, Th57;
A27: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
A28: (f2 . O) `1 = - 1 by A1, A3, A13, Th54;
A29: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
set x = the Element of (rng f2) /\ (rng g2);
A30: dom f = 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))))) . p3 = f2 . I by A14, FUNCT_1:13;
A32: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A33: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
b - a > 0 by A1, XREAL_1:50;
then A34: 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 A23, Th50;
then A35: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A36: (f . I) `2 = c by A5, A14;
A37: dom g = 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))))) . p2 = g2 . O by A15, FUNCT_1:13;
then A38: (g2 . O) `1 > (f2 . I) `1 by A11, A34, A23, A31, Th50;
A39: p3 `1 > a by A9, A10, XXREAL_0:2;
then A40: (f2 . O) `2 <= 1 by A1, A2, A7, A8, A13, A14, A24, A36, Th61;
A41: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A42: (f2 . I) `2 = - 1 by A2, A5, A14, Th57;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A37, FUNCT_1:13;
then A43: (g2 . I) `1 < (f2 . I) `1 by A10, A34, A23, A31, Th50;
- 1 <= (f2 . O) `2 by A1, A2, A7, A8, A13, A14, A39, A24, A36, Th61;
then rng f2 meets rng g2 by A29, A28, A42, A40, A41, A27, A26, A33, A25, A22, A32, A38, A43, Th33, JGRAPH_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A37, A45, A46, FUNCT_1:13;
A48: g . z2 in rng g by A37, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, XBOOLE_0:def_4;
then consider z1 being set such that
A50: z1 in dom f2 and
A51: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def_3;
A52: f . z1 in rng f by A30, A50, FUNCT_1:def_3;
f . z1 in the carrier of (TOP-REAL 2) by A50, 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 A30, A50, A51, FUNCT_1:13;
then f . z1 = g . z2 by A47, A53, A49, A35, FUNCT_1:def_4;
hence rng f meets rng g by A52, A48, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:107
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = a & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = a and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: a < p4 `1 and
A10: p4 `1 < p3 `1 and
A11: p3 `1 < p2 `1 and
A12: p2 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th106; ::_thesis: verum
end;
theorem Th108: :: JGRAPH_7:108
for p1, p2, p3, p4 being 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 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < 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
proof
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 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < 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); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < 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 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = d and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 < p4 `1 and
A11: p4 `1 <= b 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) `2 = d by A4, A14;
A22: a < p2 `1 by A7, A8, XXREAL_0:2;
p2 `1 < p4 `1 by A9, A10, XXREAL_0:2;
then A23: (g2 . I) `1 <= 1 by A1, A11, A14, A15, A22, A21, Th62;
A24: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A25: (g2 . I) `2 = 1 by A2, A6, A15, Th55;
A26: (g2 . O) `2 = 1 by A2, A4, A14, Th55;
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_1:13;
A29: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A15, A27, FUNCT_1:13;
d - c > 0 by A2, XREAL_1:50;
then A30: 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) ;
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) `2 = 1 by A2, A5, A13, Th55;
A34: (f . I) `2 = d by A5, A13;
A35: p3 `1 < b by A10, A11, XXREAL_0:2;
p1 `1 < p3 `1 by A8, A9, XXREAL_0:2;
then A36: - 1 <= (f2 . O) `1 by A1, A7, A12, A13, A35, A34, Th62;
A37: (f2 . O) `2 = 1 by A2, A3, A12, Th55;
set x = the 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_1:13;
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) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 by A8, A30, Th50;
A42: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 by A10, A40, A30, Th50;
A43: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A9, A40, A30, Th50;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A12, A38, FUNCT_1:13;
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, Th34;
then rng f2 meets rng g2 by A31, A32, A20, A24, JGRAPH_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A27, A45, A46, FUNCT_1:13;
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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, 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 A38, A49, FUNCT_1:def_3;
A52: g . z2 in rng g by A27, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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_2:5;
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;
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 A38, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A47, A54, A53, A48, FUNCT_1:def_4;
hence rng f meets rng g by A51, A52, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:109
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = d & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = d and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 < p4 `1 and
A11: p4 `1 <= b and
A12: P is_an_arc_of p1,p3 and
A13: Q is_an_arc_of p2,p4 and
A14: P c= closed_inside_of_rectangle (a,b,c,d) and
A15: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A16: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A13, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A12, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A14, A15, A16, Th108; ::_thesis: verum
end;
theorem Th110: :: JGRAPH_7:110
for p1, p2, p3, p4 being 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 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & 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
proof
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 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & 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 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & 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 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & 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 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 <= b 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 `1 < p3 `1 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 A1, A2, A6, A11, A12, A16, Th63;
A23: (g2 . I) `1 = 1 by A1, A6, A16, Th56;
A24: (g2 . O) `2 = 1 by A2, A4, A15, Th55;
(f . I) `2 = d by A5, A14;
then A25: - 1 <= (f2 . O) `1 by A1, A7, A10, A13, A14, A21, Th62;
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) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 by A8, A29, Th50;
(f . O) `2 = d by A3, A13;
then A32: (f2 . I) `1 <= 1 by A1, A7, A10, A13, A14, A21, Th62;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A30, A29, 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: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A35: (f2 . I) `2 = 1 by A2, A5, A14, Th55;
A36: (f2 . O) `2 = 1 by A2, A3, A13, Th55;
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) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A9, A30, A29, Th50;
A40: (g2 . I) `2 <= 1 by A1, A2, A6, A11, A12, A16, Th63;
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, Th35, 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
end;
theorem :: JGRAPH_7:111
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & c <= p4 `2 & p4 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 <= b and
A11: c <= p4 `2 and
A12: p4 `2 <= d and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th110; ::_thesis: verum
end;
theorem Th112: :: JGRAPH_7:112
for p1, p2, p3, p4 being 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 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < 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
proof
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 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < 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); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < 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 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 <= b and
A11: a < p4 `1 and
A12: p4 `1 <= b 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 `1 < p3 `1 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) `1 by A1, A11, A12, A16, Th64;
A23: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
A24: (g2 . O) `2 = 1 by A2, A4, A15, Th55;
(f . I) `2 = d by A5, A14;
then A25: - 1 <= (f2 . O) `1 by A1, A7, A10, A13, A14, A21, Th62;
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) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 by A8, A29, Th50;
(f . O) `2 = d by A3, A13;
then A32: (f2 . I) `1 <= 1 by A1, A7, A10, A13, A14, A21, Th62;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A30, A29, 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: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A35: (f2 . I) `2 = 1 by A2, A5, A14, Th55;
A36: (f2 . O) `2 = 1 by A2, A3, A13, Th55;
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) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A9, A30, A29, Th50;
A40: (g2 . I) `1 <= 1 by A1, A11, A12, A16, Th64;
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, Th36, 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
end;
theorem :: JGRAPH_7:113
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = d & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 < p3 `1 & p3 `1 <= b & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = d and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 < p3 `1 and
A10: p3 `1 <= b and
A11: a < p4 `1 and
A12: p4 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th112; ::_thesis: verum
end;
theorem Th114: :: JGRAPH_7:114
for p1, p2, p3, p4 being 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 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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
proof
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 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `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 `2 = d and
A4: p2 `2 = d and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: c <= p4 `2 and
A11: p4 `2 < p3 `2 and
A12: p3 `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 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) ;
A21: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
a < p2 `1 by A7, A8, XXREAL_0:2;
then A22: (g2 . O) `1 <= 1 by A1, A2, A4, A9, A15, Th63;
A23: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A24: (g2 . I) `1 = 1 by A1, A6, A16, Th56;
d - c > 0 by A2, XREAL_1:50;
then A25: 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) ;
A26: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
p3 `2 > c by A10, A11, XXREAL_0:2;
then A27: (f2 . I) `2 <= 1 by A1, A2, A5, A12, A14, Th63;
A28: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A29: (f2 . I) `1 = 1 by A1, A5, A14, Th56;
p1 `1 <= b by A8, A9, XXREAL_0:2;
then A30: - 1 <= (f2 . O) `1 by A1, A2, A3, A7, A13, Th63;
A31: (f2 . O) `2 = 1 by A2, A3, A13, Th55;
set x = the Element of (rng f2) /\ (rng g2);
A32: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A33: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A14, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A34: 2 / (b - a) > 0 by XREAL_1:139;
then A35: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 by A8, A25, Th50;
d > p4 `2 by A11, A12, XXREAL_0:2;
then A36: - 1 <= (g2 . I) `2 by A1, A2, A6, A10, A16, Th63;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A34, A25, Th51;
then A37: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A38: (g2 . O) `2 = 1 by A2, A4, A15, Th55;
A39: dom g = the carrier of I[01] by FUNCT_2:def_1;
then A40: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, FUNCT_1:13;
A41: ((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 A11, A34, A25, Th51;
A42: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A39, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A13, A32, FUNCT_1:13;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A35, A41, A40, A33, A42, A31, A29, A30, A27, A38, A24, A22, A36, Th37;
then rng f2 meets rng g2 by A26, A28, A21, A23, 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 A39, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A39, 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 A32, 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 A32, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A37, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:115
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: c <= p4 `2 and
A11: p4 `2 < p3 `2 and
A12: p3 `2 <= d and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th114; ::_thesis: verum
end;
theorem Th116: :: JGRAPH_7:116
for p1, p2, p3, p4 being 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 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & 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
proof
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 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & 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); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & 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 `2 = d and
A4: p2 `2 = d and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: c <= p3 `2 and
A11: p3 `2 <= d 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) ; ::_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) ;
A22: - 1 <= (f2 . I) `2 by A1, A2, A5, A10, A11, A15, Th63;
A23: (f2 . I) `2 <= 1 by A1, A2, A5, A10, A11, A15, Th63;
A24: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A18, Th53;
A25: (f2 . I) `1 = 1 by A1, A5, A15, Th56;
A26: (f2 . O) `2 = 1 by A2, A3, A14, Th55;
A27: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
d - c > 0 by A2, XREAL_1:50;
then A28: 2 / (d - c) > 0 by XREAL_1:139;
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) ;
A29: - 1 < (g2 . I) `1 by A1, A12, A13, A17, Th64;
A30: (g2 . O) `2 = 1 by A2, A4, A16, Th55;
A31: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A19, Th53;
a < p2 `1 by A7, A8, XXREAL_0:2;
then A32: (g2 . O) `1 <= 1 by A1, A9, A16, Th64;
A33: (g2 . I) `1 <= 1 by A1, A12, A13, A17, Th64;
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))))) . p2 = g2 . O by A16, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A36: 2 / (b - a) > 0 by XREAL_1:139;
then A37: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 by A8, A28, Th50;
p1 `1 <= b by A8, A9, XXREAL_0:2;
then A38: - 1 <= (f2 . O) `1 by A1, A2, A3, A7, A14, Th63;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A36, A28, Th51;
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: (g2 . I) `2 = - 1 by A2, A6, A17, Th57;
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 A37, A35, A24, A26, A25, A38, A22, A23, A27, A31, A30, A41, A32, A29, A33, A40, Th38, 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 A34, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A34, 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 A42, 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 A42, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A39, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:117
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p3 `2 & p3 `2 <= d & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: c <= p3 `2 and
A11: p3 `2 <= d and
A12: a < p4 `1 and
A13: p4 `1 <= b and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th116; ::_thesis: verum
end;
theorem Th118: :: JGRAPH_7:118
for p1, p2, p3, p4 being 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 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < 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
proof
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 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < 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); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < 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 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: a < p4 `1 and
A11: p4 `1 < p3 `1 and
A12: p3 `1 <= b 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: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
A22: (f2 . I) `2 = - 1 by A2, A5, A14, Th57;
A23: (f2 . O) `2 = 1 by A2, A3, A13, Th55;
d - c > 0 by A2, XREAL_1:50;
then A24: 2 / (d - c) > 0 by XREAL_1:139;
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) ;
A25: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
b >= p4 `1 by A11, A12, XXREAL_0:2;
then A26: - 1 < (g2 . I) `1 by A1, A10, A16, Th64;
A27: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A28: p3 `1 > a by A10, A11, XXREAL_0:2;
then A29: (f2 . I) `1 <= 1 by A1, A12, A14, Th64;
A30: (f . I) `2 = c by A5, A14;
p1 `1 <= b by A8, A9, XXREAL_0:2;
then A31: - 1 <= (f2 . O) `1 by A1, A7, A12, A13, A14, A28, A30, Th64;
A32: (g2 . O) `2 = 1 by A2, A4, A15, Th55;
A33: dom g = the carrier of I[01] by FUNCT_2:def_1;
then A34: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, FUNCT_1:13;
a < p2 `1 by A7, A8, XXREAL_0:2;
then A35: (g2 . O) `1 <= 1 by A1, A9, A15, Th64;
A36: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
set x = the Element of (rng f2) /\ (rng g2);
A37: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A38: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A14, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A39: 2 / (b - a) > 0 by XREAL_1:139;
then A40: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 by A8, A24, Th50;
A41: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A42: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 by A11, A39, A24, Th50;
A43: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A33, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A13, A37, FUNCT_1:13;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A40, A42, A34, A38, A43, A23, A22, A31, A29, A32, A36, A35, A26, Th39;
then rng f2 meets rng g2 by A21, A41, A25, A27, JGRAPH_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A33, A45, A46, FUNCT_1:13;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A39, A24, 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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, 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 A37, A49, FUNCT_1:def_3;
A52: g . z2 in rng g by A33, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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_2:5;
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;
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 A37, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A47, A54, A53, A48, FUNCT_1:def_4;
hence rng f meets rng g by A51, A52, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:119
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = d and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 < p2 `1 and
A9: p2 `1 <= b and
A10: a < p4 `1 and
A11: p4 `1 < p3 `1 and
A12: p3 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th118; ::_thesis: verum
end;
theorem Th120: :: JGRAPH_7:120
for p1, p2, p3, p4 being 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 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & 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
proof
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 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & 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 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & 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 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & 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 `2 = d and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: d >= p2 `2 and
A10: p2 `2 > p3 `2 and
A11: p3 `2 > p4 `2 and
A12: p4 `2 >= c 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 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) ;
A21: p2 `2 > p4 `2 by A10, A11, XXREAL_0:2;
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) ;
A22: - 1 <= (f2 . O) `1 by A1, A2, A3, A7, A8, A13, Th63;
A23: (f2 . O) `2 = 1 by A2, A3, A13, Th55;
A24: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
A25: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A26: (f2 . I) `1 = 1 by A1, A5, A14, Th56;
A27: (f2 . O) `1 <= 1 by A1, A2, A3, A7, A8, A13, Th63;
set x = the Element of (rng f2) /\ (rng g2);
A28: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A29: (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 A30: 2 / (d - c) > 0 by XREAL_1:139;
b - a > 0 by A1, XREAL_1:50;
then A31: 2 / (b - a) > 0 by XREAL_1:139;
then A32: ((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 A10, A30, Th51;
(g . O) `1 = b by A4, A15;
then A33: (g2 . I) `2 >= - 1 by A2, A9, A12, A15, A16, A21, Th65;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A31, A30, Th51;
then A34: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A35: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A36: ((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 A11, A31, A30, Th51;
A37: (g2 . I) `1 = 1 by A1, A6, A16, Th56;
A38: (g2 . O) `1 = 1 by A1, A4, A15, Th56;
(g . I) `1 = b by A6, A16;
then A39: 1 >= (g2 . O) `2 by A2, A9, A12, A15, A16, A21, Th65;
A40: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
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))))) . p4 = g2 . I by A16, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, A41, FUNCT_1:13;
then rng f2 meets rng g2 by A32, A36, A29, A42, A24, A23, A26, A22, A27, A25, A40, A38, A37, A39, A33, A35, Th40, 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 A28, 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 A28, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A34, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:121
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: d >= p2 `2 and
A10: p2 `2 > p3 `2 and
A11: p3 `2 > p4 `2 and
A12: p4 `2 >= c and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th120; ::_thesis: verum
end;
theorem Th122: :: JGRAPH_7:122
for p1, p2, p3, p4 being 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 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & 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
proof
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 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & 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); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & 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 `2 = d and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: d >= p2 `2 and
A10: p2 `2 > p3 `2 and
A11: p3 `2 >= c 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) ; ::_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) ;
A22: - 1 <= (f2 . O) `1 by A1, A2, A3, A7, A8, A14, Th63;
A23: (f2 . O) `1 <= 1 by A1, A2, A3, A7, A8, A14, Th63;
A24: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A18, Th53;
A25: (f2 . I) `1 = 1 by A1, A5, A15, Th56;
A26: (f2 . O) `2 = 1 by A2, A3, A14, Th55;
A27: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
d - c > 0 by A2, XREAL_1:50;
then A28: 2 / (d - c) > 0 by XREAL_1:139;
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) ;
A29: - 1 < (g2 . I) `1 by A1, A2, A6, A12, A13, A17, Th66;
A30: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A21, Th52;
A31: (g2 . I) `2 = - 1 by A2, A6, A17, Th57;
c < p2 `2 by A10, A11, XXREAL_0:2;
then A32: (g2 . O) `2 <= 1 by A1, A2, A4, A9, A16, Th66;
A33: (g2 . O) `1 = 1 by A1, A4, A16, Th56;
A34: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A19, Th53;
A35: (g2 . I) `1 <= 1 by A1, A2, A6, A12, A13, A17, Th66;
set x = the Element of (rng f2) /\ (rng g2);
A36: dom f = 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))))) . p3 = f2 . I by A15, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A38: 2 / (b - a) > 0 by XREAL_1:139;
then 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 A10, A28, Th51;
d >= p3 `2 by A9, A10, XXREAL_0:2;
then A40: - 1 <= (f2 . I) `2 by A1, A2, A5, A11, A15, Th63;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A38, A28, Th51;
then A41: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A42: dom g = 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))))) . p2 = g2 . O by A16, FUNCT_1:13;
then rng f2 meets rng g2 by A39, A37, A24, A26, A25, A22, A23, A40, A27, A34, A33, A31, A32, A29, A35, A30, Th41, 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 A42, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A42, 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 A36, 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 A36, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A41, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:123
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `1 = b & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & d >= p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: d >= p2 `2 and
A10: p2 `2 > p3 `2 and
A11: p3 `2 >= c and
A12: a < p4 `1 and
A13: p4 `1 <= b and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th122; ::_thesis: verum
end;
theorem Th124: :: JGRAPH_7:124
for p1, p2, p3, p4 being 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 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & 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
proof
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 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & 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); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & 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 `2 = d and
A4: p2 `1 = b and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b 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) ; ::_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) ;
A22: - 1 <= (g2 . O) `2 by A1, A2, A4, A9, A10, A16, Th66;
A23: (g2 . O) `2 <= 1 by A1, A2, A4, A9, A10, A16, Th66;
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, Th56;
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: p3 `1 > a by A11, A12, XXREAL_0:2;
then A30: (f2 . I) `1 <= 1 by A1, A13, A15, Th64;
A31: (f2 . I) `2 = - 1 by A2, A5, A15, Th57;
A32: (f2 . O) `2 = 1 by A2, A3, A14, Th55;
A33: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A18, Th53;
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;
b - a > 0 by A1, XREAL_1:50;
then A36: 2 / (b - a) > 0 by XREAL_1:139;
then A37: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 < ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A12, A28, Th50;
p4 `1 < b by A12, A13, XXREAL_0:2;
then A38: - 1 < (g2 . I) `1 by A1, A2, A6, A11, A17, Th66;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A36, 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: (f . I) `2 = c by A5, A15;
then A41: - 1 <= (f2 . O) `1 by A1, A7, A8, A13, A14, A15, A29, Th64;
A42: (f2 . O) `1 <= 1 by A1, A7, A8, A13, A14, A15, A29, A40, Th64;
A43: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
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 rng f2 meets rng g2 by A37, A35, A33, A32, A31, A41, A42, A30, A43, A24, A26, A25, A22, A23, A38, A27, Th42, JGRAPH_6:79;
then A45: (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
A46: z2 in dom g2 and
A47: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A48: 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, A46, A47, FUNCT_1:13;
A49: g . z2 in rng g by A34, A46, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A46, FUNCT_2:5;
then A50: 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 A45, XBOOLE_0:def_4;
then consider z1 being set such that
A51: z1 in dom f2 and
A52: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def_3;
A53: f . z1 in rng f by A44, A51, FUNCT_1:def_3;
f . z1 in the carrier of (TOP-REAL 2) by A51, FUNCT_2:5;
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;
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, A51, A52, FUNCT_1:13;
then f . z1 = g . z2 by A48, A54, A50, A39, FUNCT_1:def_4;
hence rng f meets rng g by A53, A49, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:125
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `1 = b & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & c <= p2 `2 & p2 `2 <= d & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `1 = b and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b 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: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle (a,b,c,d) and
A17: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A18: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A15, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A14, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A16, A17, A18, Th124; ::_thesis: verum
end;
theorem Th126: :: JGRAPH_7:126
for p1, p2, p3, p4 being 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 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `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
proof
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 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `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); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `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 `2 = d and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: a < p4 `1 and
A10: p4 `1 < p3 `1 and
A11: p3 `1 < p2 `1 and
A12: p2 `1 <= b 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 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) ;
A21: p2 `1 > p4 `1 by A10, A11, XXREAL_0:2;
(g . O) `2 = c by A4, A15;
then A22: (g2 . I) `1 > - 1 by A1, A9, A12, A15, A16, A21, Th67;
d - c > 0 by A2, XREAL_1:50;
then A23: 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) ;
A24: b > p3 `1 by A11, A12, XXREAL_0:2;
(g . I) `2 = c by A6, A16;
then A25: 1 >= (g2 . O) `1 by A1, A9, A12, A15, A16, A21, Th67;
A26: (g2 . O) `2 = - 1 by A2, A4, A15, Th57;
A27: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
A28: (f2 . O) `2 = 1 by A2, A3, A13, Th55;
A29: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
set x = the Element of (rng f2) /\ (rng g2);
A30: dom f = 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))))) . p3 = f2 . I by A14, FUNCT_1:13;
A32: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A33: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
b - a > 0 by A1, XREAL_1:50;
then A34: 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 A23, Th51;
then A35: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A36: (f . I) `2 = c by A5, A14;
A37: dom g = 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))))) . p2 = g2 . O by A15, FUNCT_1:13;
then A38: (g2 . O) `1 > (f2 . I) `1 by A11, A34, A23, A31, Th50;
A39: p3 `1 > a by A9, A10, XXREAL_0:2;
then A40: (f2 . O) `1 <= 1 by A1, A7, A8, A13, A14, A24, A36, Th64;
A41: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A42: (f2 . I) `2 = - 1 by A2, A5, A14, Th57;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A37, FUNCT_1:13;
then A43: (g2 . I) `1 < (f2 . I) `1 by A10, A34, A23, A31, Th50;
- 1 <= (f2 . O) `1 by A1, A7, A8, A13, A14, A39, A24, A36, Th64;
then rng f2 meets rng g2 by A29, A28, A42, A40, A41, A27, A26, A33, A25, A22, A32, A38, A43, Th43, JGRAPH_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A37, A45, A46, FUNCT_1:13;
A48: g . z2 in rng g by A37, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, XBOOLE_0:def_4;
then consider z1 being set such that
A50: z1 in dom f2 and
A51: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def_3;
A52: f . z1 in rng f by A30, A50, FUNCT_1:def_3;
f . z1 in the carrier of (TOP-REAL 2) by A50, 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 A30, A50, A51, FUNCT_1:13;
then f . z1 = g . z2 by A47, A53, A49, A35, FUNCT_1:def_4;
hence rng f meets rng g by A52, A48, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:127
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = c & p3 `2 = c & p4 `2 = c & a <= p1 `1 & p1 `1 <= b & a < p4 `1 & p4 `1 < p3 `1 & p3 `1 < p2 `1 & p2 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = d and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: a <= p1 `1 and
A8: p1 `1 <= b and
A9: a < p4 `1 and
A10: p4 `1 < p3 `1 and
A11: p3 `1 < p2 `1 and
A12: p2 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th126; ::_thesis: verum
end;
theorem Th128: :: JGRAPH_7:128
for p1, p2, p3, p4 being 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 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & 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
proof
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 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & 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 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & 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 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & 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 = b and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 > p3 `2 and
A10: p3 `2 > p4 `2 and
A11: p4 `2 >= c 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 = b by A4, A14;
A22: d > 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, Th65;
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, Th56;
A26: (g2 . O) `1 = 1 by A1, A4, A14, Th56;
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_1:13;
A29: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A15, A27, FUNCT_1:13;
d - c > 0 by A2, XREAL_1:50;
then A30: 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) ;
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, Th56;
A34: (f . I) `1 = b by A5, A13;
A35: p3 `2 > c 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, Th65;
A37: (f2 . O) `1 = 1 by A1, A3, A12, Th56;
set x = the 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_1:13;
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, 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_1:13;
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, Th44;
then rng f2 meets rng g2 by A31, A32, A20, A24, JGRAPH_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A27, A45, A46, FUNCT_1:13;
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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, 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 A38, A49, FUNCT_1:def_3;
A52: g . z2 in rng g by A27, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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_2:5;
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;
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 A38, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A47, A54, A53, A48, FUNCT_1:def_4;
hence rng f meets rng g by A51, A52, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:129
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `1 = b & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 > p4 `2 & p4 `2 >= c & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `1 = b and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 > p3 `2 and
A10: p3 `2 > p4 `2 and
A11: p4 `2 >= c and
A12: P is_an_arc_of p1,p3 and
A13: Q is_an_arc_of p2,p4 and
A14: P c= closed_inside_of_rectangle (a,b,c,d) and
A15: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A16: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A13, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A12, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A14, A15, A16, Th128; ::_thesis: verum
end;
theorem Th130: :: JGRAPH_7:130
for p1, p2, p3, p4 being 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 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & 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
proof
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 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & 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 ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & 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); ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & 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 = b and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 > p3 `2 and
A10: p3 `2 >= c and
A11: a < p4 `1 and
A12: p4 `1 <= b 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) `1 by A1, A2, A6, A11, A12, A16, Th66;
A23: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
A24: (g2 . O) `1 = 1 by A1, A4, A15, Th56;
(f . I) `1 = b by A5, A14;
then A25: 1 >= (f2 . O) `2 by A2, A7, A10, A13, A14, A21, Th65;
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 = b by A3, A13;
then A32: (f2 . I) `2 >= - 1 by A2, A7, A10, A13, A14, A21, Th65;
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, Th56;
A36: (f2 . O) `1 = 1 by A1, A3, A13, Th56;
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) `1 <= 1 by A1, A2, A6, A11, A12, A16, Th66;
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, Th45, 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
end;
theorem :: JGRAPH_7:131
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `1 = b & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 > p3 `2 & p3 `2 >= c & a < p4 `1 & p4 `1 <= b & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `1 = b and
A5: p3 `1 = b and
A6: p4 `2 = c and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 > p3 `2 and
A10: p3 `2 >= c and
A11: a < p4 `1 and
A12: p4 `1 <= b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th130; ::_thesis: verum
end;
theorem Th132: :: JGRAPH_7:132
for p1, p2, p3, p4 being 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 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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
proof
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 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 = b and
A4: p2 `1 = b and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 >= c and
A10: b >= p3 `1 and
A11: p3 `1 > p4 `1 and
A12: p4 `1 > a 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 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) ;
A21: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
d > p2 `2 by A7, A8, XXREAL_0:2;
then A22: - 1 <= (g2 . O) `2 by A1, A2, A4, A9, A15, Th66;
A23: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A24: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
d - c > 0 by A2, XREAL_1:50;
then A25: 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) ;
A26: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
p3 `1 > a by A11, A12, XXREAL_0:2;
then A27: (f2 . I) `1 <= 1 by A1, A2, A5, A10, A14, Th66;
A28: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A29: (f2 . I) `2 = - 1 by A2, A5, A14, Th57;
p1 `2 >= c by A8, A9, XXREAL_0:2;
then A30: (f2 . O) `2 <= 1 by A1, A2, A3, A7, A13, Th66;
A31: (f2 . O) `1 = 1 by A1, A3, A13, Th56;
set x = the Element of (rng f2) /\ (rng g2);
A32: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A33: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A14, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A34: 2 / (b - a) > 0 by XREAL_1:139;
then A35: ((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, A25, Th51;
b > p4 `1 by A10, A11, XXREAL_0:2;
then A36: - 1 < (g2 . I) `1 by A1, A2, A6, A12, A16, Th66;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A34, A25, Th51;
then A37: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A38: (g2 . O) `1 = 1 by A1, A4, A15, Th56;
A39: dom g = the carrier of I[01] by FUNCT_2:def_1;
then A40: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, FUNCT_1:13;
A41: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 by A11, A34, A25, Th50;
A42: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A39, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A13, A32, FUNCT_1:13;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A35, A41, A40, A33, A42, A31, A29, A30, A27, A38, A24, A22, A36, Th46;
then rng f2 meets rng g2 by A26, A28, A21, A23, 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 A39, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A39, 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 A32, 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 A32, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A37, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:133
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `1 = b and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 >= c and
A10: b >= p3 `1 and
A11: p3 `1 > p4 `1 and
A12: p4 `1 > a and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th132; ::_thesis: verum
end;
theorem Th134: :: JGRAPH_7:134
for p1, p2, p3, p4 being 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 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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
proof
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 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 = b and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: b >= p2 `1 and
A10: p2 `1 > p3 `1 and
A11: p3 `1 > p4 `1 and
A12: p4 `1 > a 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 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) ;
A21: p2 `1 > p4 `1 by A10, A11, XXREAL_0:2;
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) ;
A22: - 1 <= (f2 . O) `2 by A1, A2, A3, A7, A8, A13, Th66;
A23: (f2 . O) `1 = 1 by A1, A3, A13, Th56;
A24: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
A25: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A26: (f2 . I) `2 = - 1 by A2, A5, A14, Th57;
A27: (f2 . O) `2 <= 1 by A1, A2, A3, A7, A8, A13, Th66;
set x = the Element of (rng f2) /\ (rng g2);
A28: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A29: (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 A30: 2 / (d - c) > 0 by XREAL_1:139;
b - a > 0 by A1, XREAL_1:50;
then A31: 2 / (b - a) > 0 by XREAL_1:139;
then A32: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A10, A30, Th50;
(g . O) `2 = c by A4, A15;
then A33: (g2 . I) `1 > - 1 by A1, A9, A12, A15, A16, A21, Th67;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A31, A30, Th50;
then A34: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def_5;
A35: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A36: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 by A11, A31, A30, Th50;
A37: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
A38: (g2 . O) `2 = - 1 by A2, A4, A15, Th57;
(g . I) `2 = c by A6, A16;
then A39: 1 >= (g2 . O) `1 by A1, A9, A12, A15, A16, A21, Th67;
A40: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
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))))) . p4 = g2 . I by A16, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, A41, FUNCT_1:13;
then rng f2 meets rng g2 by A32, A36, A29, A42, A24, A23, A26, A22, A27, A25, A40, A38, A37, A39, A33, A35, Th47, 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 A28, 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 A28, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A34, FUNCT_1:def_4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:135
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `1 = b & p2 `2 = c & p3 `2 = c & p4 `2 = c & c <= p1 `2 & p1 `2 <= d & b >= p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: c <= p1 `2 and
A8: p1 `2 <= d and
A9: b >= p2 `1 and
A10: p2 `1 > p3 `1 and
A11: p3 `1 > p4 `1 and
A12: p4 `1 > a and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle (a,b,c,d) and
A16: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A17: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A14, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A13, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A15, A16, A17, Th134; ::_thesis: verum
end;
theorem Th136: :: JGRAPH_7:136
for p1, p2, p3, p4 being 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 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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
proof
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 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 `2 = c and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: b >= p1 `1 and
A8: p1 `1 > p2 `1 and
A9: p2 `1 > p3 `1 and
A10: p3 `1 > p4 `1 and
A11: p4 `1 > a 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) `2 = c by A4, A14;
A22: b > p2 `1 by A7, A8, XXREAL_0:2;
p2 `1 > p4 `1 by A9, A10, XXREAL_0:2;
then A23: (g2 . I) `1 > - 1 by A1, A11, A14, A15, A22, A21, Th67;
A24: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A25: (g2 . I) `2 = - 1 by A2, A6, A15, Th57;
A26: (g2 . O) `2 = - 1 by A2, A4, A14, Th57;
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_1:13;
A29: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A15, A27, FUNCT_1:13;
d - c > 0 by A2, XREAL_1:50;
then A30: 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) ;
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) `2 = - 1 by A2, A5, A13, Th57;
A34: (f . I) `2 = c by A5, A13;
A35: p3 `1 > a by A10, A11, XXREAL_0:2;
p1 `1 > p3 `1 by A8, A9, XXREAL_0:2;
then A36: 1 >= (f2 . O) `1 by A1, A7, A12, A13, A35, A34, Th67;
A37: (f2 . O) `2 = - 1 by A2, A3, A12, Th57;
set x = the 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_1:13;
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) `1 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 by A8, A30, Th50;
A42: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4) `1 by A10, A40, A30, Th50;
A43: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `1 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3) `1 by A9, A40, A30, Th50;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A12, A38, FUNCT_1:13;
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, Th48;
then rng f2 meets rng g2 by A31, A32, A20, A24, JGRAPH_6:79;
then A44: (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
A45: z2 in dom g2 and
A46: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def_3;
A47: 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 A27, A45, A46, FUNCT_1:13;
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;
the Element of (rng f2) /\ (rng g2) in rng f2 by A44, 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 A38, A49, FUNCT_1:def_3;
A52: g . z2 in rng g by A27, A45, FUNCT_1:def_3;
g . z2 in the carrier of (TOP-REAL 2) by A45, FUNCT_2:5;
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_2:5;
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;
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 A38, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A47, A54, A53, A48, FUNCT_1:def_4;
hence rng f meets rng g by A51, A52, XBOOLE_0:3; ::_thesis: verum
end;
theorem :: JGRAPH_7:137
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number
for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let a, b, c, d be real number ; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st a < b & c < d & p1 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) holds
P meets Q
let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 `2 = c & p2 `2 = c & p3 `2 = c & p4 `2 = c & b >= p1 `1 & p1 `1 > p2 `1 & p2 `1 > p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle (a,b,c,d) & Q c= closed_inside_of_rectangle (a,b,c,d) implies P meets Q )
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = c and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: b >= p1 `1 and
A8: p1 `1 > p2 `1 and
A9: p2 `1 > p3 `1 and
A10: p3 `1 > p4 `1 and
A11: p4 `1 > a and
A12: P is_an_arc_of p1,p3 and
A13: Q is_an_arc_of p2,p4 and
A14: P c= closed_inside_of_rectangle (a,b,c,d) and
A15: Q c= closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: P meets Q
A16: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = Q & g . 0 = p2 & g . 1 = p4 ) by A13, Th2;
ex f being Function of I[01],(TOP-REAL 2) st
( f is continuous & f is one-to-one & rng f = P & f . 0 = p1 & f . 1 = p3 ) by A12, Th2;
hence P meets Q by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A14, A15, A16, Th136; ::_thesis: verum
end;