:: 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;