:: JGRAPH_6 semantic presentation begin Lm1: for a, b being real number st b <= 0 & a <= b holds a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) proof let a, b be real number ; ::_thesis: ( b <= 0 & a <= b implies a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) ) assume that A1: b <= 0 and A2: a <= b ; ::_thesis: a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) A3: a ^2 >= 0 by XREAL_1:63; then A4: (- b) * (sqrt (1 + (a ^2))) = sqrt (((- b) ^2) * (1 + (a ^2))) by A1, SQUARE_1:54; A5: (- a) * (sqrt (1 + (b ^2))) = sqrt (((- a) ^2) * (1 + (b ^2))) by A1, A2, SQUARE_1:54; ( a < b or a = b ) by A2, XXREAL_0:1; then ( b ^2 < a ^2 or a = b ) by A1, SQUARE_1:44; then A6: ((b ^2) * 1) + ((b ^2) * (a ^2)) <= ((a ^2) * 1) + ((a ^2) * (b ^2)) by XREAL_1:7; b ^2 >= 0 by XREAL_1:63; then - (a * (sqrt (1 + (b ^2)))) >= - (b * (sqrt (1 + (a ^2)))) by A3, A4, A5, A6, SQUARE_1:26; hence a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) by XREAL_1:24; ::_thesis: verum end; Lm2: for a, b being real number st a <= 0 & a <= b holds a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) proof let a, b be real number ; ::_thesis: ( a <= 0 & a <= b implies a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) ) assume that A1: a <= 0 and A2: a <= b ; ::_thesis: a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) now__::_thesis:_(_(_b_<=_0_&_a_*_(sqrt_(1_+_(b_^2)))_<=_b_*_(sqrt_(1_+_(a_^2)))_)_or_(_b_>_0_&_a_*_(sqrt_(1_+_(b_^2)))_<=_b_*_(sqrt_(1_+_(a_^2)))_)_) percases ( b <= 0 or b > 0 ) ; case b <= 0 ; ::_thesis: a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) hence a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) by A2, Lm1; ::_thesis: verum end; caseA3: b > 0 ; ::_thesis: a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) b ^2 >= 0 by XREAL_1:63; then sqrt (1 + (b ^2)) > 0 by SQUARE_1:25; then A4: a * (sqrt (1 + (b ^2))) <= 0 by A1; a ^2 >= 0 by XREAL_1:63; then sqrt (1 + (a ^2)) > 0 by SQUARE_1:25; hence a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) by A3, A4; ::_thesis: verum end; end; end; hence a * (sqrt (1 + (b ^2))) <= b * (sqrt (1 + (a ^2))) ; ::_thesis: verum end; Lm3: for a, b being real number st a >= 0 & a >= b holds a * (sqrt (1 + (b ^2))) >= b * (sqrt (1 + (a ^2))) proof let a, b be real number ; ::_thesis: ( a >= 0 & a >= b implies a * (sqrt (1 + (b ^2))) >= b * (sqrt (1 + (a ^2))) ) assume that A1: a >= 0 and A2: a >= b ; ::_thesis: a * (sqrt (1 + (b ^2))) >= b * (sqrt (1 + (a ^2))) - a <= - b by A2, XREAL_1:24; then (- a) * (sqrt (1 + ((- b) ^2))) <= (- b) * (sqrt (1 + ((- a) ^2))) by A1, Lm2; then - (a * (sqrt (1 + (b ^2)))) <= - (b * (sqrt (1 + (a ^2)))) ; hence a * (sqrt (1 + (b ^2))) >= b * (sqrt (1 + (a ^2))) by XREAL_1:24; ::_thesis: verum end; theorem Th1: :: JGRAPH_6:1 for a, c, d being real number for p being Point of (TOP-REAL 2) st c <= d & p in LSeg (|[a,c]|,|[a,d]|) holds ( p `1 = a & c <= p `2 & p `2 <= d ) proof let a, c, d be real number ; ::_thesis: for p being Point of (TOP-REAL 2) st c <= d & p in LSeg (|[a,c]|,|[a,d]|) holds ( p `1 = a & c <= p `2 & p `2 <= d ) let p be Point of (TOP-REAL 2); ::_thesis: ( c <= d & p in LSeg (|[a,c]|,|[a,d]|) implies ( p `1 = a & c <= p `2 & p `2 <= d ) ) assume that A1: c <= d and A2: p in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: ( p `1 = a & c <= p `2 & p `2 <= d ) thus p `1 = a by A2, TOPREAL3:11; ::_thesis: ( c <= p `2 & p `2 <= d ) A3: |[a,c]| `2 = c by EUCLID:52; |[a,d]| `2 = d by EUCLID:52; hence ( c <= p `2 & p `2 <= d ) by A1, A2, A3, TOPREAL1:4; ::_thesis: verum end; theorem Th2: :: JGRAPH_6:2 for a, c, d being real number for p being Point of (TOP-REAL 2) st c < d & p `1 = a & c <= p `2 & p `2 <= d holds p in LSeg (|[a,c]|,|[a,d]|) proof let a, c, d be real number ; ::_thesis: for p being Point of (TOP-REAL 2) st c < d & p `1 = a & c <= p `2 & p `2 <= d holds p in LSeg (|[a,c]|,|[a,d]|) let p be Point of (TOP-REAL 2); ::_thesis: ( c < d & p `1 = a & c <= p `2 & p `2 <= d implies p in LSeg (|[a,c]|,|[a,d]|) ) assume that A1: c < d and A2: p `1 = a and A3: c <= p `2 and A4: p `2 <= d ; ::_thesis: p in LSeg (|[a,c]|,|[a,d]|) A5: d - c > 0 by A1, XREAL_1:50; reconsider w = ((p `2) - c) / (d - c) as Real ; A6: ((1 - w) * |[a,c]|) + (w * |[a,d]|) = |[((1 - w) * a),((1 - w) * c)]| + (w * |[a,d]|) by EUCLID:58 .= |[((1 - w) * a),((1 - w) * c)]| + |[(w * a),(w * d)]| by EUCLID:58 .= |[(((1 - w) * a) + (w * a)),(((1 - w) * c) + (w * d))]| by EUCLID:56 .= |[a,(c + (w * (d - c)))]| .= |[a,(c + ((p `2) - c))]| by A5, XCMPLX_1:87 .= p by A2, EUCLID:53 ; A7: (p `2) - c >= 0 by A3, XREAL_1:48; (p `2) - c <= d - c by A4, XREAL_1:9; then w <= (d - c) / (d - c) by A5, XREAL_1:72; then w <= 1 by A5, XCMPLX_1:60; hence p in LSeg (|[a,c]|,|[a,d]|) by A5, A6, A7; ::_thesis: verum end; theorem Th3: :: JGRAPH_6:3 for a, b, d being real number for p being Point of (TOP-REAL 2) st a <= b & p in LSeg (|[a,d]|,|[b,d]|) holds ( p `2 = d & a <= p `1 & p `1 <= b ) proof let a, b, d be real number ; ::_thesis: for p being Point of (TOP-REAL 2) st a <= b & p in LSeg (|[a,d]|,|[b,d]|) holds ( p `2 = d & a <= p `1 & p `1 <= b ) let p be Point of (TOP-REAL 2); ::_thesis: ( a <= b & p in LSeg (|[a,d]|,|[b,d]|) implies ( p `2 = d & a <= p `1 & p `1 <= b ) ) assume that A1: a <= b and A2: p in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: ( p `2 = d & a <= p `1 & p `1 <= b ) thus p `2 = d by A2, TOPREAL3:12; ::_thesis: ( a <= p `1 & p `1 <= b ) A3: |[a,d]| `1 = a by EUCLID:52; |[b,d]| `1 = b by EUCLID:52; hence ( a <= p `1 & p `1 <= b ) by A1, A2, A3, TOPREAL1:3; ::_thesis: verum end; theorem Th4: :: JGRAPH_6:4 for a, b being real number for B being Subset of I[01] st B = [.a,b.] holds B is closed proof let a, b be real number ; ::_thesis: for B being Subset of I[01] st B = [.a,b.] holds B is closed let B be Subset of I[01]; ::_thesis: ( B = [.a,b.] implies B is closed ) assume A1: B = [.a,b.] ; ::_thesis: B is closed reconsider B2 = B as Subset of R^1 by BORSUK_1:40, TOPMETR:17, XBOOLE_1:1; A2: B2 is closed by A1, TREAL_1:1; reconsider I1 = [.0,1.] as Subset of R^1 by TOPMETR:17; A3: [#] (R^1 | I1) = the carrier of I[01] by BORSUK_1:40, PRE_TOPC:def_5; A4: I[01] = R^1 | I1 by TOPMETR:19, TOPMETR:20; B = B2 /\ ([#] (R^1 | I1)) by A3, XBOOLE_1:28; hence B is closed by A2, A4, PRE_TOPC:13; ::_thesis: verum end; theorem Th5: :: JGRAPH_6:5 for X being TopStruct for Y, Z being non empty TopStruct for f being Function of X,Y for g being Function of X,Z holds ( dom f = dom g & dom f = the carrier of X & dom f = [#] X ) proof let X be TopStruct ; ::_thesis: for Y, Z being non empty TopStruct for f being Function of X,Y for g being Function of X,Z holds ( dom f = dom g & dom f = the carrier of X & dom f = [#] X ) let Y, Z be non empty TopStruct ; ::_thesis: for f being Function of X,Y for g being Function of X,Z holds ( dom f = dom g & dom f = the carrier of X & dom f = [#] X ) let f be Function of X,Y; ::_thesis: for g being Function of X,Z holds ( dom f = dom g & dom f = the carrier of X & dom f = [#] X ) let g be Function of X,Z; ::_thesis: ( dom f = dom g & dom f = the carrier of X & dom f = [#] X ) dom f = the carrier of X by FUNCT_2:def_1; hence ( dom f = dom g & dom f = the carrier of X & dom f = [#] X ) by FUNCT_2:def_1; ::_thesis: verum end; theorem Th6: :: JGRAPH_6:6 for X being non empty TopSpace for B being non empty Subset of X ex f being Function of (X | B),X st ( ( for p being Point of (X | B) holds f . p = p ) & f is continuous ) proof let X be non empty TopSpace; ::_thesis: for B being non empty Subset of X ex f being Function of (X | B),X st ( ( for p being Point of (X | B) holds f . p = p ) & f is continuous ) let B be non empty Subset of X; ::_thesis: ex f being Function of (X | B),X st ( ( for p being Point of (X | B) holds f . p = p ) & f is continuous ) defpred S1[ set , set ] means for p being Point of (X | B) holds $2 = $1; A1: [#] (X | B) = B by PRE_TOPC:def_5; A2: for x being Element of (X | B) ex y being Element of X st S1[x,y] proof let x be Element of (X | B); ::_thesis: ex y being Element of X st S1[x,y] x in B by A1; then reconsider px = x as Point of X ; set y0 = px; S1[x,px] ; hence ex y being Element of X st S1[x,y] ; ::_thesis: verum end; ex g being Function of the carrier of (X | B), the carrier of X st for x being Element of (X | B) holds S1[x,g . x] from FUNCT_2:sch_3(A2); then consider g being Function of the carrier of (X | B), the carrier of X such that A3: for x being Element of (X | B) holds S1[x,g . x] ; A4: for p being Point of (X | B) holds g . p = p by A3; A5: for r0 being Point of (X | B) for V being Subset of X st g . r0 in V & V is open holds ex W being Subset of (X | B) st ( r0 in W & W is open & g .: W c= V ) proof let r0 be Point of (X | B); ::_thesis: for V being Subset of X st g . r0 in V & V is open holds ex W being Subset of (X | B) st ( r0 in W & W is open & g .: W c= V ) let V be Subset of X; ::_thesis: ( g . r0 in V & V is open implies ex W being Subset of (X | B) st ( r0 in W & W is open & g .: W c= V ) ) assume that A6: g . r0 in V and A7: V is open ; ::_thesis: ex W being Subset of (X | B) st ( r0 in W & W is open & g .: W c= V ) reconsider W2 = V /\ ([#] (X | B)) as Subset of (X | B) ; g . r0 = r0 by A3; then A8: r0 in W2 by A6, XBOOLE_0:def_4; A9: W2 is open by A7, TOPS_2:24; g .: W2 c= V proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in g .: W2 or y in V ) assume y in g .: W2 ; ::_thesis: y in V then consider x being set such that A10: x in dom g and A11: x in W2 and A12: y = g . x by FUNCT_1:def_6; reconsider px = x as Point of (X | B) by A10; g . px = px by A3; hence y in V by A11, A12, XBOOLE_0:def_4; ::_thesis: verum end; hence ex W being Subset of (X | B) st ( r0 in W & W is open & g .: W c= V ) by A8, A9; ::_thesis: verum end; reconsider g1 = g as Function of (X | B),X ; g1 is continuous by A5, JGRAPH_2:10; hence ex f being Function of (X | B),X st ( ( for p being Point of (X | B) holds f . p = p ) & f is continuous ) by A4; ::_thesis: verum end; theorem Th7: :: JGRAPH_6:7 for X being non empty TopSpace for f1 being Function of X,R^1 for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 - a ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 - a ) & g is continuous ) let f1 be Function of X,R^1; ::_thesis: for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 - a ) & g is continuous ) let a be real number ; ::_thesis: ( f1 is continuous implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 - a ) & g is continuous ) ) assume f1 is continuous ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 - a ) & g is continuous ) then consider g1 being Function of X,R^1 such that A1: for p being Point of X for r1 being real number st f1 . p = r1 holds g1 . p = r1 + (- a) and A2: g1 is continuous by JGRAPH_2:24; for p being Point of X for r1 being real number st f1 . p = r1 holds g1 . p = r1 - a proof let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds g1 . p = r1 - a let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g1 . p = r1 - a ) assume f1 . p = r1 ; ::_thesis: g1 . p = r1 - a then g1 . p = r1 + (- a) by A1; hence g1 . p = r1 - a ; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = r1 - a ) & g is continuous ) by A2; ::_thesis: verum end; theorem Th8: :: JGRAPH_6:8 for X being non empty TopSpace for f1 being Function of X,R^1 for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a - r1 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a - r1 ) & g is continuous ) let f1 be Function of X,R^1; ::_thesis: for a being real number st f1 is continuous holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a - r1 ) & g is continuous ) let a be real number ; ::_thesis: ( f1 is continuous implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a - r1 ) & g is continuous ) ) assume f1 is continuous ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a - r1 ) & g is continuous ) then consider g1 being Function of X,R^1 such that A1: for p being Point of X for r1 being real number st f1 . p = r1 holds g1 . p = r1 - a and A2: g1 is continuous by Th7; consider g2 being Function of X,R^1 such that A3: for p being Point of X for r1 being real number st g1 . p = r1 holds g2 . p = - r1 and A4: g2 is continuous by A2, JGRAPH_4:8; for p being Point of X for r1 being real number st f1 . p = r1 holds g2 . p = a - r1 proof let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds g2 . p = a - r1 let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g2 . p = a - r1 ) assume f1 . p = r1 ; ::_thesis: g2 . p = a - r1 then g1 . p = r1 - a by A1; then g2 . p = - (r1 - a) by A3 .= a - r1 ; hence g2 . p = a - r1 ; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1 being real number st f1 . p = r1 holds g . p = a - r1 ) & g is continuous ) by A4; ::_thesis: verum end; theorem Th9: :: JGRAPH_6:9 for X being non empty TopSpace for n being Element of NAT for p being Point of (TOP-REAL n) for f being Function of X,R^1 st f is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (f . r) * p ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for n being Element of NAT for p being Point of (TOP-REAL n) for f being Function of X,R^1 st f is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (f . r) * p ) & g is continuous ) let n be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL n) for f being Function of X,R^1 st f is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (f . r) * p ) & g is continuous ) let p be Point of (TOP-REAL n); ::_thesis: for f being Function of X,R^1 st f is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (f . r) * p ) & g is continuous ) let f be Function of X,R^1; ::_thesis: ( f is continuous implies ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (f . r) * p ) & g is continuous ) ) assume A1: f is continuous ; ::_thesis: ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (f . r) * p ) & g is continuous ) defpred S1[ set , set ] means $2 = (f . $1) * p; A2: for x being Element of X ex y being Element of (TOP-REAL n) st S1[x,y] ; ex g being Function of the carrier of X, the carrier of (TOP-REAL n) st for x being Element of X holds S1[x,g . x] from FUNCT_2:sch_3(A2); then consider g being Function of the carrier of X, the carrier of (TOP-REAL n) such that A3: for x being Element of X holds S1[x,g . x] ; reconsider g = g as Function of X,(TOP-REAL n) ; for r0 being Point of X for V being Subset of (TOP-REAL n) st g . r0 in V & V is open holds ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) proof let r0 be Point of X; ::_thesis: for V being Subset of (TOP-REAL n) st g . r0 in V & V is open holds ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) let V be Subset of (TOP-REAL n); ::_thesis: ( g . r0 in V & V is open implies ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) ) assume that A4: g . r0 in V and A5: V is open ; ::_thesis: ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) A6: g . r0 in Int V by A4, A5, TOPS_1:23; reconsider u = g . r0 as Point of (Euclid n) by TOPREAL3:8; consider s being real number such that A7: s > 0 and A8: Ball (u,s) c= V by A6, GOBOARD6:5; now__::_thesis:_(_(_p_<>_0._(TOP-REAL_n)_&_ex_W_being_Subset_of_X_st_ (_r0_in_W_&_W_is_open_&_g_.:_W_c=_V_)_)_or_(_p_=_0._(TOP-REAL_n)_&_ex_W_being_Subset_of_X_st_ (_r0_in_W_&_W_is_open_&_g_.:_W_c=_V_)_)_) percases ( p <> 0. (TOP-REAL n) or p = 0. (TOP-REAL n) ) ; caseA9: p <> 0. (TOP-REAL n) ; ::_thesis: ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) then A10: |.p.| <> 0 by TOPRNS_1:24; set r2 = s / |.p.|; reconsider G1 = ].((f . r0) - (s / |.p.|)),((f . r0) + (s / |.p.|)).[ as Subset of R^1 by TOPMETR:17; A11: f . r0 < (f . r0) + (s / |.p.|) by A7, A10, XREAL_1:29, XREAL_1:139; then (f . r0) - (s / |.p.|) < f . r0 by XREAL_1:19; then A12: f . r0 in G1 by A11, XXREAL_1:4; G1 is open by JORDAN6:35; then consider W2 being Subset of X such that A13: r0 in W2 and A14: W2 is open and A15: f .: W2 c= G1 by A1, A12, JGRAPH_2:10; g .: W2 c= V proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in g .: W2 or y in V ) assume y in g .: W2 ; ::_thesis: y in V then consider r being set such that A16: r in dom g and A17: r in W2 and A18: y = g . r by FUNCT_1:def_6; reconsider r = r as Point of X by A16; dom f = the carrier of X by FUNCT_2:def_1; then f . r in f .: W2 by A17, FUNCT_1:def_6; then A19: abs ((f . r) - (f . r0)) < s / |.p.| by A15, RCOMP_1:1; reconsider t = f . r, t0 = f . r0 as Real by XREAL_0:def_1; A20: abs (t0 - t) = abs (t - t0) by UNIFORM1:11; reconsider v = g . r as Point of (Euclid n) by TOPREAL3:8; g . r0 = (f . r0) * p by A3; then A21: |.((g . r0) - (g . r)).| = |.(((f . r0) * p) - ((f . r) * p)).| by A3 .= |.(((f . r0) - (f . r)) * p).| by EUCLID:50 .= (abs (t0 - t)) * |.p.| by TOPRNS_1:7 ; (abs ((f . r) - (f . r0))) * |.p.| < (s / |.p.|) * |.p.| by A10, A19, XREAL_1:68; then |.((g . r0) - (g . r)).| < s by A9, A20, A21, TOPRNS_1:24, XCMPLX_1:87; then dist (u,v) < s by JGRAPH_1:28; then g . r in Ball (u,s) by METRIC_1:11; hence y in V by A8, A18; ::_thesis: verum end; hence ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) by A13, A14; ::_thesis: verum end; caseA22: p = 0. (TOP-REAL n) ; ::_thesis: ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) A23: for r being Point of X holds g . r = 0. (TOP-REAL n) proof let r be Point of X; ::_thesis: g . r = 0. (TOP-REAL n) thus g . r = (f . r) * p by A3 .= 0. (TOP-REAL n) by A22, EUCLID:28 ; ::_thesis: verum end; then A24: 0. (TOP-REAL n) in V by A4; set W2 = [#] X; g .: ([#] X) c= V proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in g .: ([#] X) or y in V ) assume y in g .: ([#] X) ; ::_thesis: y in V then ex x being set st ( x in dom g & x in [#] X & y = g . x ) by FUNCT_1:def_6; hence y in V by A23, A24; ::_thesis: verum end; hence ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) ; ::_thesis: verum end; end; end; hence ex W being Subset of X st ( r0 in W & W is open & g .: W c= V ) ; ::_thesis: verum end; then g is continuous by JGRAPH_2:10; hence ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (f . r) * p ) & g is continuous ) by A3; ::_thesis: verum end; theorem Th10: :: JGRAPH_6:10 Sq_Circ . |[(- 1),0]| = |[(- 1),0]| proof set p = |[(- 1),0]|; A1: |[(- 1),0]| `1 = - 1 by EUCLID:52; A2: |[(- 1),0]| `2 = 0 by EUCLID:52; A3: |[(- 1),0]| <> 0. (TOP-REAL 2) by A1, EUCLID:52, EUCLID:54; |[(- 1),0]| `2 <= - (|[(- 1),0]| `1) by A1, EUCLID:52; then Sq_Circ . |[(- 1),0]| = |[((|[(- 1),0]| `1) / (sqrt (1 + (((|[(- 1),0]| `2) / (|[(- 1),0]| `1)) ^2)))),((|[(- 1),0]| `2) / (sqrt (1 + (((|[(- 1),0]| `2) / (|[(- 1),0]| `1)) ^2))))]| by A1, A2, A3, JGRAPH_3:def_1; hence Sq_Circ . |[(- 1),0]| = |[(- 1),0]| by A2, EUCLID:52, SQUARE_1:18; ::_thesis: verum end; theorem :: JGRAPH_6:11 for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } holds Sq_Circ . |[(- 1),0]| = W-min P by Th10, JGRAPH_5:29; theorem Th12: :: JGRAPH_6:12 for X being non empty TopSpace for n being Element of NAT for g1, g2 being Function of X,(TOP-REAL n) st g1 is continuous & g2 is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (g1 . r) + (g2 . r) ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for n being Element of NAT for g1, g2 being Function of X,(TOP-REAL n) st g1 is continuous & g2 is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (g1 . r) + (g2 . r) ) & g is continuous ) let n be Element of NAT ; ::_thesis: for g1, g2 being Function of X,(TOP-REAL n) st g1 is continuous & g2 is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (g1 . r) + (g2 . r) ) & g is continuous ) let g1, g2 be Function of X,(TOP-REAL n); ::_thesis: ( g1 is continuous & g2 is continuous implies ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (g1 . r) + (g2 . r) ) & g is continuous ) ) assume that A1: g1 is continuous and A2: g2 is continuous ; ::_thesis: ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (g1 . r) + (g2 . r) ) & g is continuous ) defpred S1[ set , set ] means for r1, r2 being Element of (TOP-REAL n) st g1 . $1 = r1 & g2 . $1 = r2 holds $2 = r1 + r2; A3: for x being Element of X ex y being Element of (TOP-REAL n) st S1[x,y] proof let x be Element of X; ::_thesis: ex y being Element of (TOP-REAL n) st S1[x,y] set rr1 = g1 . x; set rr2 = g2 . x; set r3 = (g1 . x) + (g2 . x); for s1, s2 being Point of (TOP-REAL n) st g1 . x = s1 & g2 . x = s2 holds (g1 . x) + (g2 . x) = s1 + s2 ; hence ex y being Element of (TOP-REAL n) st S1[x,y] ; ::_thesis: verum end; ex f being Function of the carrier of X, the carrier of (TOP-REAL n) st for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A3); then consider f being Function of the carrier of X, the carrier of (TOP-REAL n) such that A4: for x being Element of X for r1, r2 being Element of (TOP-REAL n) st g1 . x = r1 & g2 . x = r2 holds f . x = r1 + r2 ; reconsider g0 = f as Function of X,(TOP-REAL n) ; A5: for r being Point of X holds g0 . r = (g1 . r) + (g2 . r) by A4; for p being Point of X for V being Subset of (TOP-REAL n) st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) proof let p be Point of X; ::_thesis: for V being Subset of (TOP-REAL n) st g0 . p in V & V is open holds ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) let V be Subset of (TOP-REAL n); ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) ) assume that A6: g0 . p in V and A7: V is open ; ::_thesis: ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) A8: g0 . p in Int V by A6, A7, TOPS_1:23; reconsider r = g0 . p as Point of (Euclid n) by TOPREAL3:8; consider r0 being real number such that A9: r0 > 0 and A10: Ball (r,r0) c= V by A8, GOBOARD6:5; reconsider r01 = g1 . p as Point of (Euclid n) by TOPREAL3:8; reconsider G1 = Ball (r01,(r0 / 2)) as Subset of (TOP-REAL n) by TOPREAL3:8; reconsider r02 = g2 . p as Point of (Euclid n) by TOPREAL3:8; reconsider G2 = Ball (r02,(r0 / 2)) as Subset of (TOP-REAL n) by TOPREAL3:8; A11: g1 . p in G1 by A9, GOBOARD6:1, XREAL_1:215; A12: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; then reconsider GG1 = G1, GG2 = G2 as Subset of (TopSpaceMetr (Euclid n)) ; GG1 is open by TOPMETR:14; then G1 is open by A12, PRE_TOPC:30; then consider W1 being Subset of X such that A13: p in W1 and A14: W1 is open and A15: g1 .: W1 c= G1 by A1, A11, JGRAPH_2:10; A16: g2 . p in G2 by A9, GOBOARD6:1, XREAL_1:215; GG2 is open by TOPMETR:14; then G2 is open by A12, PRE_TOPC:30; then consider W2 being Subset of X such that A17: p in W2 and A18: W2 is open and A19: g2 .: W2 c= G2 by A2, A16, JGRAPH_2:10; set W = W1 /\ W2; A20: p in W1 /\ W2 by A13, A17, XBOOLE_0:def_4; g0 .: (W1 /\ W2) c= Ball (r,r0) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: (W1 /\ W2) or x in Ball (r,r0) ) assume x in g0 .: (W1 /\ W2) ; ::_thesis: x in Ball (r,r0) then consider z being set such that A21: z in dom g0 and A22: z in W1 /\ W2 and A23: g0 . z = x by FUNCT_1:def_6; A24: z in W1 by A22, XBOOLE_0:def_4; reconsider pz = z as Point of X by A21; dom g1 = the carrier of X by FUNCT_2:def_1; then A25: g1 . pz in g1 .: W1 by A24, FUNCT_1:def_6; reconsider aa1 = g1 . pz as Point of (TOP-REAL n) ; reconsider bb1 = aa1 as Point of (Euclid n) by TOPREAL3:8; dist (r01,bb1) < r0 / 2 by A15, A25, METRIC_1:11; then A26: |.((g1 . p) - (g1 . pz)).| < r0 / 2 by JGRAPH_1:28; A27: z in W2 by A22, XBOOLE_0:def_4; dom g2 = the carrier of X by FUNCT_2:def_1; then A28: g2 . pz in g2 .: W2 by A27, FUNCT_1:def_6; reconsider aa2 = g2 . pz as Point of (TOP-REAL n) ; reconsider bb2 = aa2 as Point of (Euclid n) by TOPREAL3:8; dist (r02,bb2) < r0 / 2 by A19, A28, METRIC_1:11; then A29: |.((g2 . p) - (g2 . pz)).| < r0 / 2 by JGRAPH_1:28; A30: aa1 + aa2 = x by A4, A23; reconsider bb0 = aa1 + aa2 as Point of (Euclid n) by TOPREAL3:8; A31: g0 . pz = (g1 . pz) + (g2 . pz) by A4; ((g1 . p) + (g2 . p)) - ((g1 . pz) + (g2 . pz)) = (((g1 . p) + (g2 . p)) - (g1 . pz)) - (g2 . pz) by EUCLID:46 .= (((g1 . p) + (g2 . p)) + (- (g1 . pz))) - (g2 . pz) by EUCLID:41 .= (((g1 . p) + (g2 . p)) + (- (g1 . pz))) + (- (g2 . pz)) by EUCLID:41 .= (((g1 . p) + (- (g1 . pz))) + (g2 . p)) + (- (g2 . pz)) by EUCLID:26 .= ((g1 . p) + (- (g1 . pz))) + ((g2 . p) + (- (g2 . pz))) by EUCLID:26 .= ((g1 . p) - (g1 . pz)) + ((g2 . p) + (- (g2 . pz))) by EUCLID:41 .= ((g1 . p) - (g1 . pz)) + ((g2 . p) - (g2 . pz)) by EUCLID:41 ; then A32: |.(((g1 . p) + (g2 . p)) - ((g1 . pz) + (g2 . pz))).| <= |.((g1 . p) - (g1 . pz)).| + |.((g2 . p) - (g2 . pz)).| by TOPRNS_1:29; |.((g1 . p) - (g1 . pz)).| + |.((g2 . p) - (g2 . pz)).| < (r0 / 2) + (r0 / 2) by A26, A29, XREAL_1:8; then |.(((g1 . p) + (g2 . p)) - ((g1 . pz) + (g2 . pz))).| < r0 by A32, XXREAL_0:2; then |.((g0 . p) - (g0 . pz)).| < r0 by A4, A31; then dist (r,bb0) < r0 by A23, A30, JGRAPH_1:28; hence x in Ball (r,r0) by A30, METRIC_1:11; ::_thesis: verum end; hence ex W being Subset of X st ( p in W & W is open & g0 .: W c= V ) by A10, A14, A18, A20, XBOOLE_1:1; ::_thesis: verum end; then g0 is continuous by JGRAPH_2:10; hence ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = (g1 . r) + (g2 . r) ) & g is continuous ) by A5; ::_thesis: verum end; theorem Th13: :: JGRAPH_6:13 for X being non empty TopSpace for n being Element of NAT for p1, p2 being Point of (TOP-REAL n) for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = ((f1 . r) * p1) + ((f2 . r) * p2) ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for n being Element of NAT for p1, p2 being Point of (TOP-REAL n) for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = ((f1 . r) * p1) + ((f2 . r) * p2) ) & g is continuous ) let n be Element of NAT ; ::_thesis: for p1, p2 being Point of (TOP-REAL n) for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = ((f1 . r) * p1) + ((f2 . r) * p2) ) & g is continuous ) let p1, p2 be Point of (TOP-REAL n); ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = ((f1 . r) * p1) + ((f2 . r) * p2) ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous implies ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = ((f1 . r) * p1) + ((f2 . r) * p2) ) & g is continuous ) ) assume that A1: f1 is continuous and A2: f2 is continuous ; ::_thesis: ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = ((f1 . r) * p1) + ((f2 . r) * p2) ) & g is continuous ) consider g1 being Function of X,(TOP-REAL n) such that A3: for r being Point of X holds g1 . r = (f1 . r) * p1 and A4: g1 is continuous by A1, Th9; consider g2 being Function of X,(TOP-REAL n) such that A5: for r being Point of X holds g2 . r = (f2 . r) * p2 and A6: g2 is continuous by A2, Th9; consider g being Function of X,(TOP-REAL n) such that A7: for r being Point of X holds g . r = (g1 . r) + (g2 . r) and A8: g is continuous by A4, A6, Th12; for r being Point of X holds g . r = ((f1 . r) * p1) + ((f2 . r) * p2) proof let r be Point of X; ::_thesis: g . r = ((f1 . r) * p1) + ((f2 . r) * p2) g . r = (g1 . r) + (g2 . r) by A7; then g . r = (g1 . r) + ((f2 . r) * p2) by A5; hence g . r = ((f1 . r) * p1) + ((f2 . r) * p2) by A3; ::_thesis: verum end; hence ex g being Function of X,(TOP-REAL n) st ( ( for r being Point of X holds g . r = ((f1 . r) * p1) + ((f2 . r) * p2) ) & g is continuous ) by A8; ::_thesis: verum end; begin Lm4: |[(- 1),0]| `1 = - 1 by EUCLID:52; Lm5: |[(- 1),0]| `2 = 0 by EUCLID:52; Lm6: |[1,0]| `1 = 1 by EUCLID:52; Lm7: |[1,0]| `2 = 0 by EUCLID:52; Lm8: |[0,(- 1)]| `1 = 0 by EUCLID:52; Lm9: |[0,(- 1)]| `2 = - 1 by EUCLID:52; Lm10: |[0,1]| `1 = 0 by EUCLID:52; Lm11: |[0,1]| `2 = 1 by EUCLID:52; Lm12: now__::_thesis:_(_|.|[(-_1),0]|.|_=_1_&_|.|[1,0]|.|_=_1_&_|.|[0,(-_1)]|.|_=_1_&_|.|[0,1]|.|_=_1_) thus |.|[(- 1),0]|.| = sqrt (((- 1) ^2) + (0 ^2)) by Lm4, Lm5, JGRAPH_3:1 .= 1 by SQUARE_1:18 ; ::_thesis: ( |.|[1,0]|.| = 1 & |.|[0,(- 1)]|.| = 1 & |.|[0,1]|.| = 1 ) thus |.|[1,0]|.| = sqrt ((1 ^2) + (0 ^2)) by Lm6, Lm7, JGRAPH_3:1 .= 1 by SQUARE_1:18 ; ::_thesis: ( |.|[0,(- 1)]|.| = 1 & |.|[0,1]|.| = 1 ) thus |.|[0,(- 1)]|.| = sqrt ((0 ^2) + ((- 1) ^2)) by Lm8, Lm9, JGRAPH_3:1 .= 1 by SQUARE_1:18 ; ::_thesis: |.|[0,1]|.| = 1 thus |.|[0,1]|.| = sqrt ((0 ^2) + (1 ^2)) by Lm10, Lm11, JGRAPH_3:1 .= 1 by SQUARE_1:18 ; ::_thesis: verum end; Lm13: 0 in [.0,1.] by XXREAL_1:1; Lm14: 1 in [.0,1.] by XXREAL_1:1; theorem Th14: :: JGRAPH_6:14 for f, g being Function of I[01],(TOP-REAL 2) for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2) for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds rng f meets rng g proof let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2) for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds rng f meets rng g let C0, KXP, KXN, KYP, KYN be Subset of (TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds rng f meets rng g let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 implies rng f meets rng g ) assume A1: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g then ex f2 being Function of I[01],(TOP-REAL 2) st ( f2 . 0 = f . 1 & f2 . 1 = f . 0 & rng f2 = rng f & f2 is continuous & f2 is one-to-one ) by JGRAPH_5:12; hence rng f meets rng g by A1, JGRAPH_5:13; ::_thesis: verum end; theorem Th15: :: JGRAPH_6:15 for f, g being Function of I[01],(TOP-REAL 2) for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2) for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 holds rng f meets rng g proof let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2) for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 holds rng f meets rng g let C0, KXP, KXN, KYP, KYN be Subset of (TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 holds rng f meets rng g let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 implies rng f meets rng g ) assume A1: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXP & f . I in KXN & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g then ex g2 being Function of I[01],(TOP-REAL 2) st ( g2 . 0 = g . 1 & g2 . 1 = g . 0 & rng g2 = rng g & g2 is continuous & g2 is one-to-one ) by JGRAPH_5:12; hence rng f meets rng g by A1, Th14; ::_thesis: verum end; theorem Th16: :: JGRAPH_6:16 for p1, p2, p3, p4 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g proof let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g let C0 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g ) assume that A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and A2: LE p1,p2,P and A3: LE p2,p3,P and A4: LE p3,p4,P ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 implies rng f meets rng g ) assume that A5: ( f is continuous & f is one-to-one ) and A6: ( g is continuous & g is one-to-one ) and A7: C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } and A8: f . 0 = p3 and A9: f . 1 = p1 and A10: g . 0 = p2 and A11: g . 1 = p4 and A12: rng f c= C0 and A13: rng g c= C0 ; ::_thesis: rng f meets rng g A14: dom f = the carrier of I[01] by FUNCT_2:def_1; A15: dom g = the carrier of I[01] by FUNCT_2:def_1; percases ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 or ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ) ; supposeA16: ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 ) ; ::_thesis: rng f meets rng g now__::_thesis:_(_(_p1_=_p2_&_rng_f_meets_rng_g_)_or_(_p2_=_p3_&_rng_f_meets_rng_g_)_or_(_p3_=_p4_&_rng_f_meets_rng_g_)_) percases ( p1 = p2 or p2 = p3 or p3 = p4 ) by A16; caseA17: p1 = p2 ; ::_thesis: rng f meets rng g A18: p1 in rng f by A9, A14, Lm14, BORSUK_1:40, FUNCT_1:def_3; p2 in rng g by A10, A15, Lm13, BORSUK_1:40, FUNCT_1:def_3; hence rng f meets rng g by A17, A18, XBOOLE_0:3; ::_thesis: verum end; caseA19: p2 = p3 ; ::_thesis: rng f meets rng g A20: p3 in rng f by A8, A14, Lm13, BORSUK_1:40, FUNCT_1:def_3; p2 in rng g by A10, A15, Lm13, BORSUK_1:40, FUNCT_1:def_3; hence rng f meets rng g by A19, A20, XBOOLE_0:3; ::_thesis: verum end; caseA21: p3 = p4 ; ::_thesis: rng f meets rng g A22: p3 in rng f by A8, A14, Lm13, BORSUK_1:40, FUNCT_1:def_3; p4 in rng g by A11, A15, Lm14, BORSUK_1:40, FUNCT_1:def_3; hence rng f meets rng g by A21, A22, XBOOLE_0:3; ::_thesis: verum end; end; end; hence rng f meets rng g ; ::_thesis: verum end; suppose ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ; ::_thesis: rng f meets rng g then consider h being Function of (TOP-REAL 2),(TOP-REAL 2) such that A23: h is being_homeomorphism and A24: for q being Point of (TOP-REAL 2) holds |.(h . q).| = |.q.| and A25: |[(- 1),0]| = h . p1 and A26: |[0,1]| = h . p2 and A27: |[1,0]| = h . p3 and A28: |[0,(- 1)]| = h . p4 by A1, A2, A3, A4, JGRAPH_5:67; A29: h is one-to-one by A23, TOPS_2:def_5; reconsider f2 = h * f as Function of I[01],(TOP-REAL 2) ; reconsider g2 = h * g as Function of I[01],(TOP-REAL 2) ; A30: dom f2 = the carrier of I[01] by FUNCT_2:def_1; A31: dom g2 = the carrier of I[01] by FUNCT_2:def_1; A32: f2 . 1 = |[(- 1),0]| by A9, A25, A30, Lm14, BORSUK_1:40, FUNCT_1:12; A33: g2 . 1 = |[0,(- 1)]| by A11, A28, A31, Lm14, BORSUK_1:40, FUNCT_1:12; A34: f2 . 0 = |[1,0]| by A8, A27, A30, Lm13, BORSUK_1:40, FUNCT_1:12; A35: g2 . 0 = |[0,1]| by A10, A26, A31, Lm13, BORSUK_1:40, FUNCT_1:12; A36: ( f2 is continuous & f2 is one-to-one ) by A5, A23, JGRAPH_5:5, JGRAPH_5:6; A37: ( g2 is continuous & g2 is one-to-one ) by A6, A23, JGRAPH_5:5, JGRAPH_5:6; A38: rng f2 c= C0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in C0 ) assume y in rng f2 ; ::_thesis: y in C0 then consider x being set such that A39: x in dom f2 and A40: y = f2 . x by FUNCT_1:def_3; A41: f2 . x = h . (f . x) by A39, FUNCT_1:12; A42: f . x in rng f by A14, A39, FUNCT_1:def_3; then A43: f . x in C0 by A12; reconsider qf = f . x as Point of (TOP-REAL 2) by A42; A44: ex q5 being Point of (TOP-REAL 2) st ( q5 = f . x & |.q5.| <= 1 ) by A7, A43; |.(h . qf).| = |.qf.| by A24; hence y in C0 by A7, A40, A41, A44; ::_thesis: verum end; A45: rng g2 c= C0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g2 or y in C0 ) assume y in rng g2 ; ::_thesis: y in C0 then consider x being set such that A46: x in dom g2 and A47: y = g2 . x by FUNCT_1:def_3; A48: g2 . x = h . (g . x) by A46, FUNCT_1:12; A49: g . x in rng g by A15, A46, FUNCT_1:def_3; then A50: g . x in C0 by A13; reconsider qg = g . x as Point of (TOP-REAL 2) by A49; A51: ex q5 being Point of (TOP-REAL 2) st ( q5 = g . x & |.q5.| <= 1 ) by A7, A50; |.(h . qg).| = |.qg.| by A24; hence y in C0 by A7, A47, A48, A51; ::_thesis: verum end; defpred S1[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 >= - ($1 `1) ); { q1 where q1 is Point of (TOP-REAL 2) : S1[q1] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); then reconsider KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } as Subset of (TOP-REAL 2) ; defpred S2[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ); { q2 where q2 is Point of (TOP-REAL 2) : S2[q2] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); then reconsider KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } as Subset of (TOP-REAL 2) ; defpred S3[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 >= - ($1 `1) ); { q3 where q3 is Point of (TOP-REAL 2) : S3[q3] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); then reconsider KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } as Subset of (TOP-REAL 2) ; defpred S4[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 <= - ($1 `1) ); { q4 where q4 is Point of (TOP-REAL 2) : S4[q4] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); then reconsider KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } as Subset of (TOP-REAL 2) ; reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1; - (|[(- 1),0]| `1) = 1 by Lm4; then A52: f2 . I in KXN by A32, Lm5, Lm12; A53: f2 . O in KXP by A34, Lm6, Lm7, Lm12; - (|[0,(- 1)]| `1) = 0 by Lm8; then A54: g2 . I in KYN by A33, Lm9, Lm12; - (|[0,1]| `1) = 0 by Lm10; then g2 . O in KYP by A35, Lm11, Lm12; then rng f2 meets rng g2 by A7, A36, A37, A38, A45, A52, A53, A54, Th14; then consider x2 being set such that A55: x2 in rng f2 and A56: x2 in rng g2 by XBOOLE_0:3; consider z2 being set such that A57: z2 in dom f2 and A58: x2 = f2 . z2 by A55, FUNCT_1:def_3; consider z3 being set such that A59: z3 in dom g2 and A60: x2 = g2 . z3 by A56, FUNCT_1:def_3; A61: dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A62: g . z3 in rng g by A15, A59, FUNCT_1:def_3; A63: f . z2 in rng f by A14, A57, FUNCT_1:def_3; reconsider h1 = h as Function ; A64: (h1 ") . x2 = (h1 ") . (h . (f . z2)) by A57, A58, FUNCT_1:12 .= f . z2 by A29, A61, A63, FUNCT_1:34 ; A65: (h1 ") . x2 = (h1 ") . (h . (g . z3)) by A59, A60, FUNCT_1:12 .= g . z3 by A29, A61, A62, FUNCT_1:34 ; A66: (h1 ") . x2 in rng f by A14, A57, A64, FUNCT_1:def_3; (h1 ") . x2 in rng g by A15, A59, A65, FUNCT_1:def_3; hence rng f meets rng g by A66, XBOOLE_0:3; ::_thesis: verum end; end; end; theorem Th17: :: JGRAPH_6:17 for p1, p2, p3, p4 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds rng f meets rng g proof let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds rng f meets rng g let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds rng f meets rng g let C0 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds rng f meets rng g ) assume that A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and A2: LE p1,p2,P and A3: LE p2,p3,P and A4: LE p3,p4,P ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds rng f meets rng g let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p3 & f . 1 = p1 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 implies rng f meets rng g ) assume that A5: ( f is continuous & f is one-to-one ) and A6: ( g is continuous & g is one-to-one ) and A7: C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } and A8: f . 0 = p3 and A9: f . 1 = p1 and A10: g . 0 = p4 and A11: g . 1 = p2 and A12: rng f c= C0 and A13: rng g c= C0 ; ::_thesis: rng f meets rng g A14: dom f = the carrier of I[01] by FUNCT_2:def_1; A15: dom g = the carrier of I[01] by FUNCT_2:def_1; percases ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 or ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ) ; supposeA16: ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 ) ; ::_thesis: rng f meets rng g now__::_thesis:_(_(_p1_=_p2_&_rng_f_meets_rng_g_)_or_(_p2_=_p3_&_rng_f_meets_rng_g_)_or_(_p3_=_p4_&_rng_f_meets_rng_g_)_) percases ( p1 = p2 or p2 = p3 or p3 = p4 ) by A16; caseA17: p1 = p2 ; ::_thesis: rng f meets rng g A18: p1 in rng f by A9, A14, Lm14, BORSUK_1:40, FUNCT_1:def_3; p2 in rng g by A11, A15, Lm14, BORSUK_1:40, FUNCT_1:def_3; hence rng f meets rng g by A17, A18, XBOOLE_0:3; ::_thesis: verum end; caseA19: p2 = p3 ; ::_thesis: rng f meets rng g A20: p3 in rng f by A8, A14, Lm13, BORSUK_1:40, FUNCT_1:def_3; p2 in rng g by A11, A15, Lm14, BORSUK_1:40, FUNCT_1:def_3; hence rng f meets rng g by A19, A20, XBOOLE_0:3; ::_thesis: verum end; caseA21: p3 = p4 ; ::_thesis: rng f meets rng g A22: p3 in rng f by A8, A14, Lm13, BORSUK_1:40, FUNCT_1:def_3; p4 in rng g by A10, A15, Lm13, BORSUK_1:40, FUNCT_1:def_3; hence rng f meets rng g by A21, A22, XBOOLE_0:3; ::_thesis: verum end; end; end; hence rng f meets rng g ; ::_thesis: verum end; suppose ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ; ::_thesis: rng f meets rng g then consider h being Function of (TOP-REAL 2),(TOP-REAL 2) such that A23: h is being_homeomorphism and A24: for q being Point of (TOP-REAL 2) holds |.(h . q).| = |.q.| and A25: |[(- 1),0]| = h . p1 and A26: |[0,1]| = h . p2 and A27: |[1,0]| = h . p3 and A28: |[0,(- 1)]| = h . p4 by A1, A2, A3, A4, JGRAPH_5:67; A29: h is one-to-one by A23, TOPS_2:def_5; reconsider f2 = h * f as Function of I[01],(TOP-REAL 2) ; reconsider g2 = h * g as Function of I[01],(TOP-REAL 2) ; A30: dom f2 = the carrier of I[01] by FUNCT_2:def_1; A31: dom g2 = the carrier of I[01] by FUNCT_2:def_1; A32: f2 . 1 = |[(- 1),0]| by A9, A25, A30, Lm14, BORSUK_1:40, FUNCT_1:12; A33: g2 . 1 = |[0,1]| by A11, A26, A31, Lm14, BORSUK_1:40, FUNCT_1:12; A34: f2 . 0 = |[1,0]| by A8, A27, A30, Lm13, BORSUK_1:40, FUNCT_1:12; A35: g2 . 0 = |[0,(- 1)]| by A10, A28, A31, Lm13, BORSUK_1:40, FUNCT_1:12; A36: ( f2 is continuous & f2 is one-to-one ) by A5, A23, JGRAPH_5:5, JGRAPH_5:6; A37: ( g2 is continuous & g2 is one-to-one ) by A6, A23, JGRAPH_5:5, JGRAPH_5:6; A38: rng f2 c= C0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in C0 ) assume y in rng f2 ; ::_thesis: y in C0 then consider x being set such that A39: x in dom f2 and A40: y = f2 . x by FUNCT_1:def_3; A41: f2 . x = h . (f . x) by A39, FUNCT_1:12; A42: f . x in rng f by A14, A39, FUNCT_1:def_3; then A43: f . x in C0 by A12; reconsider qf = f . x as Point of (TOP-REAL 2) by A42; A44: ex q5 being Point of (TOP-REAL 2) st ( q5 = f . x & |.q5.| <= 1 ) by A7, A43; |.(h . qf).| = |.qf.| by A24; hence y in C0 by A7, A40, A41, A44; ::_thesis: verum end; A45: rng g2 c= C0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g2 or y in C0 ) assume y in rng g2 ; ::_thesis: y in C0 then consider x being set such that A46: x in dom g2 and A47: y = g2 . x by FUNCT_1:def_3; A48: g2 . x = h . (g . x) by A46, FUNCT_1:12; A49: g . x in rng g by A15, A46, FUNCT_1:def_3; then A50: g . x in C0 by A13; reconsider qg = g . x as Point of (TOP-REAL 2) by A49; A51: ex q5 being Point of (TOP-REAL 2) st ( q5 = g . x & |.q5.| <= 1 ) by A7, A50; |.(h . qg).| = |.qg.| by A24; hence y in C0 by A7, A47, A48, A51; ::_thesis: verum end; defpred S1[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 >= - ($1 `1) ); { q1 where q1 is Point of (TOP-REAL 2) : S1[q1] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); then reconsider KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } as Subset of (TOP-REAL 2) ; defpred S2[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ); { q2 where q2 is Point of (TOP-REAL 2) : S2[q2] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); then reconsider KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } as Subset of (TOP-REAL 2) ; defpred S3[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 >= - ($1 `1) ); { q3 where q3 is Point of (TOP-REAL 2) : S3[q3] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); then reconsider KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } as Subset of (TOP-REAL 2) ; defpred S4[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 <= - ($1 `1) ); { q4 where q4 is Point of (TOP-REAL 2) : S4[q4] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); then reconsider KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } as Subset of (TOP-REAL 2) ; reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1; - (|[(- 1),0]| `1) = 1 by Lm4; then A52: f2 . I in KXN by A32, Lm5, Lm12; A53: f2 . O in KXP by A34, Lm6, Lm7, Lm12; - (|[0,(- 1)]| `1) = 0 by Lm8; then A54: g2 . I in KYP by A33, Lm10, Lm11, Lm12; - (|[0,1]| `1) = 0 by Lm10; then g2 . O in KYN by A35, Lm8, Lm9, Lm12; then rng f2 meets rng g2 by A7, A36, A37, A38, A45, A52, A53, A54, Th15; then consider x2 being set such that A55: x2 in rng f2 and A56: x2 in rng g2 by XBOOLE_0:3; consider z2 being set such that A57: z2 in dom f2 and A58: x2 = f2 . z2 by A55, FUNCT_1:def_3; consider z3 being set such that A59: z3 in dom g2 and A60: x2 = g2 . z3 by A56, FUNCT_1:def_3; A61: dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A62: g . z3 in rng g by A15, A59, FUNCT_1:def_3; A63: f . z2 in rng f by A14, A57, FUNCT_1:def_3; reconsider h1 = h as Function ; A64: (h1 ") . x2 = (h1 ") . (h . (f . z2)) by A57, A58, FUNCT_1:12 .= f . z2 by A29, A61, A63, FUNCT_1:34 ; A65: (h1 ") . x2 = (h1 ") . (h . (g . z3)) by A59, A60, FUNCT_1:12 .= g . z3 by A29, A61, A62, FUNCT_1:34 ; A66: (h1 ") . x2 in rng f by A14, A57, A64, FUNCT_1:def_3; (h1 ") . x2 in rng g by A15, A59, A65, FUNCT_1:def_3; hence rng f meets rng g by A66, XBOOLE_0:3; ::_thesis: verum end; end; end; theorem Th18: :: JGRAPH_6:18 for p1, p2, p3, p4 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1,p2,p3,p4 are_in_this_order_on P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g proof let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1,p2,p3,p4 are_in_this_order_on P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1,p2,p3,p4 are_in_this_order_on P holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g let C0 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1,p2,p3,p4 are_in_this_order_on P implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g ) assume that A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and A2: p1,p2,p3,p4 are_in_this_order_on P ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g percases ( ( LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ) or ( LE p2,p3,P & LE p3,p4,P & LE p4,p1,P ) or ( LE p3,p4,P & LE p4,p1,P & LE p1,p2,P ) or ( LE p4,p1,P & LE p1,p2,P & LE p2,p3,P ) ) by A2, JORDAN17:def_1; suppose ( LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g hence for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g by A1, JGRAPH_5:68; ::_thesis: verum end; suppose ( LE p2,p3,P & LE p3,p4,P & LE p4,p1,P ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g hence for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g by A1, JGRAPH_5:69; ::_thesis: verum end; suppose ( LE p3,p4,P & LE p4,p1,P & LE p1,p2,P ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g hence for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g by A1, Th17; ::_thesis: verum end; suppose ( LE p4,p1,P & LE p1,p2,P & LE p2,p3,P ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g hence for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g by A1, Th16; ::_thesis: verum end; end; end; begin notation let a, b, c, d be real number ; synonym rectangle (a,b,c,d) for [.a,b,c,d.]; end; Lm15: for a, b, c, d being real number st a <= b & c <= d holds rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } ) set X = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } ; assume that A1: a <= b and A2: c <= d ; ::_thesis: rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } A3: rectangle (a,b,c,d) = { p2 where p2 is Point of (TOP-REAL 2) : ( ( p2 `1 = a & p2 `2 <= d & p2 `2 >= c ) or ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) or ( p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) or ( p2 `1 = b & p2 `2 <= d & p2 `2 >= c ) ) } by A1, A2, SPPOL_2:54; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } c= rectangle (a,b,c,d) let x be set ; ::_thesis: ( x in rectangle (a,b,c,d) implies x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } ) assume x in rectangle (a,b,c,d) ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } then ex p2 being Point of (TOP-REAL 2) st ( x = p2 & ( ( p2 `1 = a & p2 `2 <= d & p2 `2 >= c ) or ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) or ( p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) or ( p2 `1 = b & p2 `2 <= d & p2 `2 >= c ) ) ) by A3; hence x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } ; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } or x in rectangle (a,b,c,d) ) assume x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } ; ::_thesis: x in rectangle (a,b,c,d) then ex p being Point of (TOP-REAL 2) st ( x = p & ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) ) ; hence x in rectangle (a,b,c,d) by A3; ::_thesis: verum end; theorem Th19: :: JGRAPH_6:19 for a, b, c, d being real number for p being Point of (TOP-REAL 2) st a <= b & c <= d & p in rectangle (a,b,c,d) holds ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) proof let a, b, c, d be real number ; ::_thesis: for p being Point of (TOP-REAL 2) st a <= b & c <= d & p in rectangle (a,b,c,d) holds ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) let p be Point of (TOP-REAL 2); ::_thesis: ( a <= b & c <= d & p in rectangle (a,b,c,d) implies ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) ) assume that A1: a <= b and A2: c <= d and A3: p in rectangle (a,b,c,d) ; ::_thesis: ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) p in { p2 where p2 is Point of (TOP-REAL 2) : ( ( p2 `1 = a & p2 `2 <= d & p2 `2 >= c ) or ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) or ( p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) or ( p2 `1 = b & p2 `2 <= d & p2 `2 >= c ) ) } by A1, A2, A3, SPPOL_2:54; then A4: ex p2 being Point of (TOP-REAL 2) st ( p2 = p & ( ( p2 `1 = a & p2 `2 <= d & p2 `2 >= c ) or ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) or ( p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) or ( p2 `1 = b & p2 `2 <= d & p2 `2 >= c ) ) ) ; percases ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) by A4; suppose ( p `1 = a & c <= p `2 & p `2 <= d ) ; ::_thesis: ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) hence ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) by A1; ::_thesis: verum end; suppose ( p `2 = d & a <= p `1 & p `1 <= b ) ; ::_thesis: ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) hence ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) by A2; ::_thesis: verum end; suppose ( p `1 = b & c <= p `2 & p `2 <= d ) ; ::_thesis: ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) hence ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) by A1; ::_thesis: verum end; suppose ( p `2 = c & a <= p `1 & p `1 <= b ) ; ::_thesis: ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) hence ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) by A2; ::_thesis: verum end; end; end; definition let a, b, c, d be real number ; func inside_of_rectangle (a,b,c,d) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 1 { p where p is Point of (TOP-REAL 2) : ( a < p `1 & p `1 < b & c < p `2 & p `2 < d ) } ; coherence { p where p is Point of (TOP-REAL 2) : ( a < p `1 & p `1 < b & c < p `2 & p `2 < d ) } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means ( a < $1 `1 & $1 `1 < b & c < $1 `2 & $1 `2 < d ); { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : ( a < p `1 & p `1 < b & c < p `2 & p `2 < d ) } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines inside_of_rectangle JGRAPH_6:def_1_:_ for a, b, c, d being real number holds inside_of_rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( a < p `1 & p `1 < b & c < p `2 & p `2 < d ) } ; definition let a, b, c, d be real number ; func closed_inside_of_rectangle (a,b,c,d) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 2 { p where p is Point of (TOP-REAL 2) : ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) } ; coherence { p where p is Point of (TOP-REAL 2) : ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means ( a <= $1 `1 & $1 `1 <= b & c <= $1 `2 & $1 `2 <= d ); { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines closed_inside_of_rectangle JGRAPH_6:def_2_:_ for a, b, c, d being real number holds closed_inside_of_rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) } ; definition let a, b, c, d be real number ; func outside_of_rectangle (a,b,c,d) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 3 { p where p is Point of (TOP-REAL 2) : ( not a <= p `1 or not p `1 <= b or not c <= p `2 or not p `2 <= d ) } ; coherence { p where p is Point of (TOP-REAL 2) : ( not a <= p `1 or not p `1 <= b or not c <= p `2 or not p `2 <= d ) } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means ( not a <= $1 `1 or not $1 `1 <= b or not c <= $1 `2 or not $1 `2 <= d ); { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : ( not a <= p `1 or not p `1 <= b or not c <= p `2 or not p `2 <= d ) } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines outside_of_rectangle JGRAPH_6:def_3_:_ for a, b, c, d being real number holds outside_of_rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( not a <= p `1 or not p `1 <= b or not c <= p `2 or not p `2 <= d ) } ; definition let a, b, c, d be real number ; func closed_outside_of_rectangle (a,b,c,d) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 4 { p where p is Point of (TOP-REAL 2) : ( not a < p `1 or not p `1 < b or not c < p `2 or not p `2 < d ) } ; coherence { p where p is Point of (TOP-REAL 2) : ( not a < p `1 or not p `1 < b or not c < p `2 or not p `2 < d ) } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means ( not a < $1 `1 or not $1 `1 < b or not c < $1 `2 or not $1 `2 < d ); { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : ( not a < p `1 or not p `1 < b or not c < p `2 or not p `2 < d ) } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines closed_outside_of_rectangle JGRAPH_6:def_4_:_ for a, b, c, d being real number holds closed_outside_of_rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( not a < p `1 or not p `1 < b or not c < p `2 or not p `2 < d ) } ; theorem Th20: :: JGRAPH_6:20 for a, b, r being real number for Kb, Cb being Subset of (TOP-REAL 2) st r >= 0 & Kb = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.(p2 - |[a,b]|).| = r } holds (AffineMap (r,a,r,b)) .: Kb = Cb proof let a, b, r be real number ; ::_thesis: for Kb, Cb being Subset of (TOP-REAL 2) st r >= 0 & Kb = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.(p2 - |[a,b]|).| = r } holds (AffineMap (r,a,r,b)) .: Kb = Cb let Kb, Cb be Subset of (TOP-REAL 2); ::_thesis: ( r >= 0 & Kb = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.(p2 - |[a,b]|).| = r } implies (AffineMap (r,a,r,b)) .: Kb = Cb ) assume A1: ( r >= 0 & Kb = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.(p2 - |[a,b]|).| = r } ) ; ::_thesis: (AffineMap (r,a,r,b)) .: Kb = Cb reconsider rr = r as Real by XREAL_0:def_1; A2: (AffineMap (r,a,r,b)) .: Kb c= Cb proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (AffineMap (r,a,r,b)) .: Kb or y in Cb ) assume y in (AffineMap (r,a,r,b)) .: Kb ; ::_thesis: y in Cb then consider x being set such that x in dom (AffineMap (r,a,r,b)) and A3: x in Kb and A4: y = (AffineMap (r,a,r,b)) . x by FUNCT_1:def_6; consider p being Point of (TOP-REAL 2) such that A5: x = p and A6: |.p.| = 1 by A1, A3; A7: (AffineMap (r,a,r,b)) . p = |[((r * (p `1)) + a),((r * (p `2)) + b)]| by JGRAPH_2:def_2; then reconsider q = y as Point of (TOP-REAL 2) by A4, A5; A8: q - |[a,b]| = |[(((r * (p `1)) + a) - a),(((r * (p `2)) + b) - b)]| by A4, A5, A7, EUCLID:62 .= r * |[(p `1),(p `2)]| by EUCLID:58 .= r * p by EUCLID:53 ; |.(r * p).| = (abs rr) * |.p.| by TOPRNS_1:7 .= r by A1, A6, ABSVALUE:def_1 ; hence y in Cb by A1, A8; ::_thesis: verum end; Cb c= (AffineMap (r,a,r,b)) .: Kb proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Cb or y in (AffineMap (r,a,r,b)) .: Kb ) assume y in Cb ; ::_thesis: y in (AffineMap (r,a,r,b)) .: Kb then consider p2 being Point of (TOP-REAL 2) such that A9: y = p2 and A10: |.(p2 - |[a,b]|).| = r by A1; now__::_thesis:_(_(_r_>_0_&_y_in_(AffineMap_(r,a,r,b))_.:_Kb_)_or_(_r_=_0_&_y_in_(AffineMap_(r,a,r,b))_.:_Kb_)_) percases ( r > 0 or r = 0 ) by A1; caseA11: r > 0 ; ::_thesis: y in (AffineMap (r,a,r,b)) .: Kb set p1 = (1 / r) * (p2 - |[a,b]|); |.((1 / r) * (p2 - |[a,b]|)).| = (abs (1 / rr)) * |.(p2 - |[a,b]|).| by TOPRNS_1:7 .= (1 / r) * r by A10, ABSVALUE:def_1 .= 1 by A11, XCMPLX_1:87 ; then A12: (1 / r) * (p2 - |[a,b]|) in Kb by A1; A13: (1 / r) * (p2 - |[a,b]|) = |[((1 / r) * ((p2 - |[a,b]|) `1)),((1 / r) * ((p2 - |[a,b]|) `2))]| by EUCLID:57; then A14: ((1 / r) * (p2 - |[a,b]|)) `1 = (1 / r) * ((p2 - |[a,b]|) `1) by EUCLID:52; A15: ((1 / r) * (p2 - |[a,b]|)) `2 = (1 / r) * ((p2 - |[a,b]|) `2) by A13, EUCLID:52; A16: r * (((1 / r) * (p2 - |[a,b]|)) `1) = (r * (1 / r)) * ((p2 - |[a,b]|) `1) by A14 .= 1 * ((p2 - |[a,b]|) `1) by A11, XCMPLX_1:87 .= (p2 `1) - (|[a,b]| `1) by TOPREAL3:3 .= (p2 `1) - a by EUCLID:52 ; A17: r * (((1 / r) * (p2 - |[a,b]|)) `2) = (r * (1 / r)) * ((p2 - |[a,b]|) `2) by A15 .= 1 * ((p2 - |[a,b]|) `2) by A11, XCMPLX_1:87 .= (p2 `2) - (|[a,b]| `2) by TOPREAL3:3 .= (p2 `2) - b by EUCLID:52 ; A18: (AffineMap (r,a,r,b)) . ((1 / r) * (p2 - |[a,b]|)) = |[((r * (((1 / r) * (p2 - |[a,b]|)) `1)) + a),((r * (((1 / r) * (p2 - |[a,b]|)) `2)) + b)]| by JGRAPH_2:def_2 .= p2 by A16, A17, EUCLID:53 ; dom (AffineMap (r,a,r,b)) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; hence y in (AffineMap (r,a,r,b)) .: Kb by A9, A12, A18, FUNCT_1:def_6; ::_thesis: verum end; caseA19: r = 0 ; ::_thesis: y in (AffineMap (r,a,r,b)) .: Kb set p1 = |[1,0]|; A20: |[1,0]| `1 = 1 by EUCLID:52; |[1,0]| `2 = 0 by EUCLID:52; then |.|[1,0]|.| = sqrt ((1 ^2) + (0 ^2)) by A20, JGRAPH_3:1 .= 1 by SQUARE_1:22 ; then A21: |[1,0]| in Kb by A1; A22: (AffineMap (r,a,r,b)) . |[1,0]| = |[((0 * (|[1,0]| `1)) + a),((0 * (|[1,0]| `2)) + b)]| by A19, JGRAPH_2:def_2 .= p2 by A10, A19, TOPRNS_1:28 ; dom (AffineMap (r,a,r,b)) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; hence y in (AffineMap (r,a,r,b)) .: Kb by A9, A21, A22, FUNCT_1:def_6; ::_thesis: verum end; end; end; hence y in (AffineMap (r,a,r,b)) .: Kb ; ::_thesis: verum end; hence (AffineMap (r,a,r,b)) .: Kb = Cb by A2, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th21: :: JGRAPH_6:21 for P, Q being Subset of (TOP-REAL 2) st ex f being Function of ((TOP-REAL 2) | P),((TOP-REAL 2) | Q) st f is being_homeomorphism & P is being_simple_closed_curve holds Q is being_simple_closed_curve proof let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( ex f being Function of ((TOP-REAL 2) | P),((TOP-REAL 2) | Q) st f is being_homeomorphism & P is being_simple_closed_curve implies Q is being_simple_closed_curve ) assume that A1: ex f being Function of ((TOP-REAL 2) | P),((TOP-REAL 2) | Q) st f is being_homeomorphism and A2: P is being_simple_closed_curve ; ::_thesis: Q is being_simple_closed_curve consider f being Function of ((TOP-REAL 2) | P),((TOP-REAL 2) | Q) such that A3: f is being_homeomorphism by A1; consider g being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P) such that A4: g is being_homeomorphism by A2, TOPREAL2:def_1; A5: |[1,0]| `1 = 1 by EUCLID:52; |[1,0]| `2 = 0 by EUCLID:52; then A6: |[1,0]| in R^2-unit_square by A5, TOPREAL1:14; A7: dom g = [#] ((TOP-REAL 2) | R^2-unit_square) by A4, TOPS_2:def_5; A8: rng g = [#] ((TOP-REAL 2) | P) by A4, TOPS_2:def_5; dom g = R^2-unit_square by A7, PRE_TOPC:def_5; then A9: g . |[1,0]| in rng g by A6, FUNCT_1:3; then A10: g . |[1,0]| in P by A8, PRE_TOPC:def_5; reconsider P1 = P as non empty Subset of (TOP-REAL 2) by A9; dom f = [#] ((TOP-REAL 2) | P) by A3, TOPS_2:def_5; then dom f = P by PRE_TOPC:def_5; then f . (g . |[1,0]|) in rng f by A10, FUNCT_1:3; then reconsider Q1 = Q as non empty Subset of (TOP-REAL 2) ; reconsider f1 = f as Function of ((TOP-REAL 2) | P1),((TOP-REAL 2) | Q1) ; reconsider g1 = g as Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P1) ; reconsider h = f1 * g1 as Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | Q1) ; h is being_homeomorphism by A3, A4, TOPS_2:57; hence Q is being_simple_closed_curve by TOPREAL2:def_1; ::_thesis: verum end; theorem Th22: :: JGRAPH_6:22 for P being Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds P is compact ; theorem Th23: :: JGRAPH_6:23 for a, b, r being real number for Cb being Subset of (TOP-REAL 2) st r > 0 & Cb = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| = r } holds Cb is being_simple_closed_curve proof let a, b, r be real number ; ::_thesis: for Cb being Subset of (TOP-REAL 2) st r > 0 & Cb = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| = r } holds Cb is being_simple_closed_curve let Cb be Subset of (TOP-REAL 2); ::_thesis: ( r > 0 & Cb = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| = r } implies Cb is being_simple_closed_curve ) assume that A1: r > 0 and A2: Cb = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| = r } ; ::_thesis: Cb is being_simple_closed_curve A3: |[r,0]| `1 = r by EUCLID:52; A4: |[r,0]| `2 = 0 by EUCLID:52; |.(|[(r + a),b]| - |[a,b]|).| = |.(|[(r + a),(0 + b)]| - |[a,b]|).| .= |.((|[r,0]| + |[a,b]|) - |[a,b]|).| by EUCLID:56 .= |.(|[r,0]| + (|[a,b]| - |[a,b]|)).| by EUCLID:45 .= |.(|[r,0]| + (0. (TOP-REAL 2))).| by EUCLID:42 .= |.|[r,0]|.| by EUCLID:27 .= sqrt ((r ^2) + (0 ^2)) by A3, A4, JGRAPH_3:1 .= r by A1, SQUARE_1:22 ; then |[(r + a),b]| in Cb by A2; then reconsider Cbb = Cb as non empty Subset of (TOP-REAL 2) ; set v = |[1,0]|; A5: |[1,0]| `1 = 1 by EUCLID:52; |[1,0]| `2 = 0 by EUCLID:52; then |.|[1,0]|.| = sqrt ((1 ^2) + (0 ^2)) by A5, JGRAPH_3:1 .= 1 by SQUARE_1:22 ; then A6: |[1,0]| in { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; defpred S1[ Point of (TOP-REAL 2)] means |.$1.| = 1; { q where q is Element of (TOP-REAL 2) : S1[q] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider Kb = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } as non empty Subset of (TOP-REAL 2) by A6; A7: the carrier of ((TOP-REAL 2) | Kb) = Kb by PRE_TOPC:8; set SC = AffineMap (r,a,r,b); A8: AffineMap (r,a,r,b) is one-to-one by A1, JGRAPH_2:44; A9: dom (AffineMap (r,a,r,b)) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A10: dom ((AffineMap (r,a,r,b)) | Kb) = (dom (AffineMap (r,a,r,b))) /\ Kb by RELAT_1:61 .= the carrier of ((TOP-REAL 2) | Kb) by A7, A9, XBOOLE_1:28 ; A11: rng ((AffineMap (r,a,r,b)) | Kb) c= ((AffineMap (r,a,r,b)) | Kb) .: the carrier of ((TOP-REAL 2) | Kb) proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng ((AffineMap (r,a,r,b)) | Kb) or u in ((AffineMap (r,a,r,b)) | Kb) .: the carrier of ((TOP-REAL 2) | Kb) ) assume u in rng ((AffineMap (r,a,r,b)) | Kb) ; ::_thesis: u in ((AffineMap (r,a,r,b)) | Kb) .: the carrier of ((TOP-REAL 2) | Kb) then ex z being set st ( z in dom ((AffineMap (r,a,r,b)) | Kb) & u = ((AffineMap (r,a,r,b)) | Kb) . z ) by FUNCT_1:def_3; hence u in ((AffineMap (r,a,r,b)) | Kb) .: the carrier of ((TOP-REAL 2) | Kb) by A10, FUNCT_1:def_6; ::_thesis: verum end; ((AffineMap (r,a,r,b)) | Kb) .: the carrier of ((TOP-REAL 2) | Kb) = (AffineMap (r,a,r,b)) .: Kb by A7, RELAT_1:129 .= Cb by A1, A2, Th20 .= the carrier of ((TOP-REAL 2) | Cbb) by PRE_TOPC:8 ; then reconsider f0 = (AffineMap (r,a,r,b)) | Kb as Function of ((TOP-REAL 2) | Kb),((TOP-REAL 2) | Cbb) by A10, A11, FUNCT_2:2; rng ((AffineMap (r,a,r,b)) | Kb) c= the carrier of (TOP-REAL 2) ; then reconsider f00 = f0 as Function of ((TOP-REAL 2) | Kb),(TOP-REAL 2) by A10, FUNCT_2:2; A12: rng f0 = ((AffineMap (r,a,r,b)) | Kb) .: the carrier of ((TOP-REAL 2) | Kb) by RELSET_1:22 .= (AffineMap (r,a,r,b)) .: Kb by A7, RELAT_1:129 .= Cb by A1, A2, Th20 ; A13: f0 is one-to-one by A8, FUNCT_1:52; Kb is compact by Th22, JGRAPH_3:26; then ex f1 being Function of ((TOP-REAL 2) | Kb),((TOP-REAL 2) | Cbb) st ( f00 = f1 & f1 is being_homeomorphism ) by A12, A13, JGRAPH_1:46, TOPMETR:7; hence Cb is being_simple_closed_curve by Th21, JGRAPH_3:26; ::_thesis: verum end; definition let a, b, r be real number ; func circle (a,b,r) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 5 { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| = r } ; coherence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| = r } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means |.($1 - |[a,b]|).| = r; { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| = r } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines circle JGRAPH_6:def_5_:_ for a, b, r being real number holds circle (a,b,r) = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| = r } ; registration let a, b, r be real number ; cluster circle (a,b,r) -> compact ; coherence circle (a,b,r) is compact proof set Cb = circle (a,b,r); percases ( r < 0 or r > 0 or r = 0 ) ; supposeA1: r < 0 ; ::_thesis: circle (a,b,r) is compact circle (a,b,r) = {} proof hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {} c= circle (a,b,r) let x be set ; ::_thesis: ( x in circle (a,b,r) implies x in {} ) assume x in circle (a,b,r) ; ::_thesis: x in {} then ex p being Point of (TOP-REAL 2) st ( x = p & |.(p - |[a,b]|).| = r ) ; hence x in {} by A1; ::_thesis: verum end; thus {} c= circle (a,b,r) by XBOOLE_1:2; ::_thesis: verum end; hence circle (a,b,r) is compact ; ::_thesis: verum end; suppose r > 0 ; ::_thesis: circle (a,b,r) is compact hence circle (a,b,r) is compact by Th22, Th23; ::_thesis: verum end; supposeA2: r = 0 ; ::_thesis: circle (a,b,r) is compact circle (a,b,r) = {|[a,b]|} proof hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {|[a,b]|} c= circle (a,b,r) let x be set ; ::_thesis: ( x in circle (a,b,r) implies x in {|[a,b]|} ) assume x in circle (a,b,r) ; ::_thesis: x in {|[a,b]|} then consider p being Point of (TOP-REAL 2) such that A3: x = p and A4: |.(p - |[a,b]|).| = r ; p = |[a,b]| by A2, A4, TOPRNS_1:28; hence x in {|[a,b]|} by A3, TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[a,b]|} or x in circle (a,b,r) ) assume x in {|[a,b]|} ; ::_thesis: x in circle (a,b,r) then A5: x = |[a,b]| by TARSKI:def_1; |.(|[a,b]| - |[a,b]|).| = 0 by TOPRNS_1:28; hence x in circle (a,b,r) by A2, A5; ::_thesis: verum end; hence circle (a,b,r) is compact ; ::_thesis: verum end; end; end; end; registration let a, b be real number ; let r be real non negative number ; cluster circle (a,b,r) -> non empty ; coherence not circle (a,b,r) is empty proof set Cb = circle (a,b,r); A1: |[r,0]| `1 = r by EUCLID:52; A2: |[r,0]| `2 = 0 by EUCLID:52; |.(|[(r + a),b]| - |[a,b]|).| = |.(|[(r + a),(0 + b)]| - |[a,b]|).| .= |.((|[r,0]| + |[a,b]|) - |[a,b]|).| by EUCLID:56 .= |.(|[r,0]| + (|[a,b]| - |[a,b]|)).| by EUCLID:45 .= |.(|[r,0]| + (0. (TOP-REAL 2))).| by EUCLID:42 .= |.|[r,0]|.| by EUCLID:27 .= sqrt ((r ^2) + (0 ^2)) by A1, A2, JGRAPH_3:1 .= r by SQUARE_1:22 ; then |[(r + a),b]| in circle (a,b,r) ; hence not circle (a,b,r) is empty ; ::_thesis: verum end; end; definition let a, b, r be real number ; func inside_of_circle (a,b,r) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 6 { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| < r } ; coherence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| < r } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means |.($1 - |[a,b]|).| < r; { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| < r } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines inside_of_circle JGRAPH_6:def_6_:_ for a, b, r being real number holds inside_of_circle (a,b,r) = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| < r } ; definition let a, b, r be real number ; func closed_inside_of_circle (a,b,r) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 7 { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| <= r } ; coherence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| <= r } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means |.($1 - |[a,b]|).| <= r; { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| <= r } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines closed_inside_of_circle JGRAPH_6:def_7_:_ for a, b, r being real number holds closed_inside_of_circle (a,b,r) = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| <= r } ; definition let a, b, r be real number ; func outside_of_circle (a,b,r) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 8 { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| > r } ; coherence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| > r } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means |.($1 - |[a,b]|).| > r; { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| > r } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines outside_of_circle JGRAPH_6:def_8_:_ for a, b, r being real number holds outside_of_circle (a,b,r) = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| > r } ; definition let a, b, r be real number ; func closed_outside_of_circle (a,b,r) -> Subset of (TOP-REAL 2) equals :: JGRAPH_6:def 9 { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| >= r } ; coherence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| >= r } is Subset of (TOP-REAL 2) proof defpred S1[ Point of (TOP-REAL 2)] means |.($1 - |[a,b]|).| >= r; { p where p is Point of (TOP-REAL 2) : S1[p] } c= the carrier of (TOP-REAL 2) from FRAENKEL:sch_10(); hence { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| >= r } is Subset of (TOP-REAL 2) ; ::_thesis: verum end; end; :: deftheorem defines closed_outside_of_circle JGRAPH_6:def_9_:_ for a, b, r being real number holds closed_outside_of_circle (a,b,r) = { p where p is Point of (TOP-REAL 2) : |.(p - |[a,b]|).| >= r } ; theorem Th24: :: JGRAPH_6:24 for r being real number holds ( inside_of_circle (0,0,r) = { p where p is Point of (TOP-REAL 2) : |.p.| < r } & ( r > 0 implies circle (0,0,r) = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| = r } ) & outside_of_circle (0,0,r) = { p3 where p3 is Point of (TOP-REAL 2) : |.p3.| > r } & closed_inside_of_circle (0,0,r) = { q where q is Point of (TOP-REAL 2) : |.q.| <= r } & closed_outside_of_circle (0,0,r) = { q2 where q2 is Point of (TOP-REAL 2) : |.q2.| >= r } ) proof let r be real number ; ::_thesis: ( inside_of_circle (0,0,r) = { p where p is Point of (TOP-REAL 2) : |.p.| < r } & ( r > 0 implies circle (0,0,r) = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| = r } ) & outside_of_circle (0,0,r) = { p3 where p3 is Point of (TOP-REAL 2) : |.p3.| > r } & closed_inside_of_circle (0,0,r) = { q where q is Point of (TOP-REAL 2) : |.q.| <= r } & closed_outside_of_circle (0,0,r) = { q2 where q2 is Point of (TOP-REAL 2) : |.q2.| >= r } ) defpred S1[ Point of (TOP-REAL 2)] means |.($1 - |[0,0]|).| < r; defpred S2[ Point of (TOP-REAL 2)] means |.$1.| < r; deffunc H1( set ) -> set = $1; A1: for p being Point of (TOP-REAL 2) holds ( S1[p] iff S2[p] ) by EUCLID:54, RLVECT_1:13; inside_of_circle (0,0,r) = { H1(p) where p is Point of (TOP-REAL 2) : S1[p] } .= { H1(p2) where p2 is Point of (TOP-REAL 2) : S2[p2] } from FRAENKEL:sch_3(A1) ; hence inside_of_circle (0,0,r) = { p where p is Point of (TOP-REAL 2) : |.p.| < r } ; ::_thesis: ( ( r > 0 implies circle (0,0,r) = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| = r } ) & outside_of_circle (0,0,r) = { p3 where p3 is Point of (TOP-REAL 2) : |.p3.| > r } & closed_inside_of_circle (0,0,r) = { q where q is Point of (TOP-REAL 2) : |.q.| <= r } & closed_outside_of_circle (0,0,r) = { q2 where q2 is Point of (TOP-REAL 2) : |.q2.| >= r } ) defpred S3[ Point of (TOP-REAL 2)] means |.($1 - |[0,0]|).| = r; defpred S4[ Point of (TOP-REAL 2)] means |.$1.| = r; A2: for p being Point of (TOP-REAL 2) holds ( S3[p] iff S4[p] ) by EUCLID:54, RLVECT_1:13; hereby ::_thesis: ( outside_of_circle (0,0,r) = { p3 where p3 is Point of (TOP-REAL 2) : |.p3.| > r } & closed_inside_of_circle (0,0,r) = { q where q is Point of (TOP-REAL 2) : |.q.| <= r } & closed_outside_of_circle (0,0,r) = { q2 where q2 is Point of (TOP-REAL 2) : |.q2.| >= r } ) assume r > 0 ; ::_thesis: circle (0,0,r) = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| = r } circle (0,0,r) = { H1(p) where p is Point of (TOP-REAL 2) : S3[p] } .= { H1(p2) where p2 is Point of (TOP-REAL 2) : S4[p2] } from FRAENKEL:sch_3(A2) ; hence circle (0,0,r) = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| = r } ; ::_thesis: verum end; defpred S5[ Point of (TOP-REAL 2)] means |.($1 - |[0,0]|).| > r; defpred S6[ Point of (TOP-REAL 2)] means |.$1.| > r; A3: for p being Point of (TOP-REAL 2) holds ( S5[p] iff S6[p] ) by EUCLID:54, RLVECT_1:13; outside_of_circle (0,0,r) = { H1(p) where p is Point of (TOP-REAL 2) : S5[p] } .= { H1(p2) where p2 is Point of (TOP-REAL 2) : S6[p2] } from FRAENKEL:sch_3(A3) ; hence outside_of_circle (0,0,r) = { p3 where p3 is Point of (TOP-REAL 2) : |.p3.| > r } ; ::_thesis: ( closed_inside_of_circle (0,0,r) = { q where q is Point of (TOP-REAL 2) : |.q.| <= r } & closed_outside_of_circle (0,0,r) = { q2 where q2 is Point of (TOP-REAL 2) : |.q2.| >= r } ) defpred S7[ Point of (TOP-REAL 2)] means |.($1 - |[0,0]|).| <= r; defpred S8[ Point of (TOP-REAL 2)] means |.$1.| <= r; A4: for p being Point of (TOP-REAL 2) holds ( S7[p] iff S8[p] ) by EUCLID:54, RLVECT_1:13; closed_inside_of_circle (0,0,r) = { H1(p) where p is Point of (TOP-REAL 2) : S7[p] } .= { H1(p2) where p2 is Point of (TOP-REAL 2) : S8[p2] } from FRAENKEL:sch_3(A4) ; hence closed_inside_of_circle (0,0,r) = { p where p is Point of (TOP-REAL 2) : |.p.| <= r } ; ::_thesis: closed_outside_of_circle (0,0,r) = { q2 where q2 is Point of (TOP-REAL 2) : |.q2.| >= r } defpred S9[ Point of (TOP-REAL 2)] means |.($1 - |[0,0]|).| >= r; defpred S10[ Point of (TOP-REAL 2)] means |.$1.| >= r; A5: for p being Point of (TOP-REAL 2) holds ( S9[p] iff S10[p] ) by EUCLID:54, RLVECT_1:13; closed_outside_of_circle (0,0,r) = { H1(p) where p is Point of (TOP-REAL 2) : S9[p] } .= { H1(p2) where p2 is Point of (TOP-REAL 2) : S10[p2] } from FRAENKEL:sch_3(A5) ; hence closed_outside_of_circle (0,0,r) = { q2 where q2 is Point of (TOP-REAL 2) : |.q2.| >= r } ; ::_thesis: verum end; theorem Th25: :: JGRAPH_6:25 for Kb, Cb being Subset of (TOP-REAL 2) st Kb = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| < 1 } holds Sq_Circ .: Kb = Cb proof let Kb, Cb be Subset of (TOP-REAL 2); ::_thesis: ( Kb = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| < 1 } implies Sq_Circ .: Kb = Cb ) assume A1: ( Kb = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| < 1 } ) ; ::_thesis: Sq_Circ .: Kb = Cb thus Sq_Circ .: Kb c= Cb :: according to XBOOLE_0:def_10 ::_thesis: Cb c= Sq_Circ .: Kb proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Sq_Circ .: Kb or y in Cb ) assume y in Sq_Circ .: Kb ; ::_thesis: y in Cb then consider x being set such that x in dom Sq_Circ and A2: x in Kb and A3: y = Sq_Circ . x by FUNCT_1:def_6; consider q being Point of (TOP-REAL 2) such that A4: q = x and A5: - 1 < q `1 and A6: q `1 < 1 and A7: - 1 < q `2 and A8: q `2 < 1 by A1, A2; now__::_thesis:_(_(_q_=_0._(TOP-REAL_2)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_<_1_)_)_or_(_q_<>_0._(TOP-REAL_2)_&_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_<_1_)_)_or_(_q_<>_0._(TOP-REAL_2)_&_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_<_1_)_)_) percases ( q = 0. (TOP-REAL 2) or ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) or ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; caseA9: q = 0. (TOP-REAL 2) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| < 1 ) then A10: Sq_Circ . q = q by JGRAPH_3:def_1; |.q.| = 0 by A9, TOPRNS_1:23; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| < 1 ) by A3, A4, A10; ::_thesis: verum end; caseA11: ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| < 1 ) then A12: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by JGRAPH_3:def_1; A13: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52; A14: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52; A15: 1 + (((q `2) / (q `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A16: now__::_thesis:_not_q_`1_=_0 assume A17: q `1 = 0 ; ::_thesis: contradiction then q `2 = 0 by A11; hence contradiction by A11, A17, EUCLID:53, EUCLID:54; ::_thesis: verum end; then A18: (q `1) ^2 > 0 by SQUARE_1:12; (q `1) ^2 < 1 ^2 by A5, A6, SQUARE_1:50; then A19: sqrt ((q `1) ^2) < 1 by A18, SQUARE_1:18, SQUARE_1:27; |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) by A13, A14, JGRAPH_3:1 .= (((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) by XCMPLX_1:76 .= (((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) by XCMPLX_1:76 .= (((q `1) ^2) / (1 + (((q `2) / (q `1)) ^2))) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) by A15, SQUARE_1:def_2 .= (((q `1) ^2) / (1 + (((q `2) / (q `1)) ^2))) + (((q `2) ^2) / (1 + (((q `2) / (q `1)) ^2))) by A15, SQUARE_1:def_2 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `2) / (q `1)) ^2)) by XCMPLX_1:62 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `2) ^2) / ((q `1) ^2))) by XCMPLX_1:76 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) / ((q `1) ^2)) + (((q `2) ^2) / ((q `1) ^2))) by A18, XCMPLX_1:60 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) + ((q `2) ^2)) / ((q `1) ^2)) by XCMPLX_1:62 .= ((q `1) ^2) * ((((q `1) ^2) + ((q `2) ^2)) / (((q `1) ^2) + ((q `2) ^2))) by XCMPLX_1:81 .= ((q `1) ^2) * 1 by A16, COMPLEX1:1, XCMPLX_1:60 .= (q `1) ^2 ; then |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| < 1 by A19, SQUARE_1:22; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| < 1 ) by A3, A4, A12; ::_thesis: verum end; caseA20: ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| < 1 ) then A21: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by JGRAPH_3:def_1; A22: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52; A23: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52; A24: 1 + (((q `1) / (q `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A25: q `2 <> 0 by A20; then A26: (q `2) ^2 > 0 by SQUARE_1:12; (q `2) ^2 < 1 ^2 by A7, A8, SQUARE_1:50; then A27: sqrt ((q `2) ^2) < 1 by A26, SQUARE_1:18, SQUARE_1:27; |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) by A22, A23, JGRAPH_3:1 .= (((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) by XCMPLX_1:76 .= (((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) by XCMPLX_1:76 .= (((q `1) ^2) / (1 + (((q `1) / (q `2)) ^2))) + (((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) by A24, SQUARE_1:def_2 .= (((q `1) ^2) / (1 + (((q `1) / (q `2)) ^2))) + (((q `2) ^2) / (1 + (((q `1) / (q `2)) ^2))) by A24, SQUARE_1:def_2 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `1) / (q `2)) ^2)) by XCMPLX_1:62 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `1) ^2) / ((q `2) ^2))) by XCMPLX_1:76 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) / ((q `2) ^2)) + (((q `2) ^2) / ((q `2) ^2))) by A26, XCMPLX_1:60 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) + ((q `2) ^2)) / ((q `2) ^2)) by XCMPLX_1:62 .= ((q `2) ^2) * ((((q `1) ^2) + ((q `2) ^2)) / (((q `1) ^2) + ((q `2) ^2))) by XCMPLX_1:81 .= ((q `2) ^2) * 1 by A25, COMPLEX1:1, XCMPLX_1:60 .= (q `2) ^2 ; then |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| < 1 by A27, SQUARE_1:22; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| < 1 ) by A3, A4, A21; ::_thesis: verum end; end; end; hence y in Cb by A1; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Cb or y in Sq_Circ .: Kb ) assume y in Cb ; ::_thesis: y in Sq_Circ .: Kb then consider p2 being Point of (TOP-REAL 2) such that A28: p2 = y and A29: |.p2.| < 1 by A1; set q = p2; now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_) percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; caseA30: p2 = 0. (TOP-REAL 2) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) then A31: p2 `1 = 0 by EUCLID:52, EUCLID:54; p2 `2 = 0 by A30, EUCLID:52, EUCLID:54; then A32: y in Kb by A1, A28, A31; A33: (Sq_Circ ") . y = y by A28, A30, JGRAPH_3:28; A34: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; y = Sq_Circ . y by A28, A33, FUNCT_1:35, JGRAPH_3:43; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A32, A34; ::_thesis: verum end; caseA35: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) set px = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]|; A36: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52; A37: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52; 1 + (((p2 `2) / (p2 `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; then A38: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by SQUARE_1:25; A39: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A40: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) = (p2 `2) / (p2 `1) by A36, A37, A38, XCMPLX_1:91; A41: p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A38, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ; A42: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A38, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ; A43: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; A44: |.p2.| ^2 < 1 ^2 by A29, SQUARE_1:16; A45: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_=_0_) assume that A46: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 and A47: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = 0 ; ::_thesis: contradiction A48: (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A46, EUCLID:52; A49: (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A47, EUCLID:52; A50: p2 `1 = 0 by A38, A48, XCMPLX_1:6; p2 `2 = 0 by A38, A49, XCMPLX_1:6; hence contradiction by A35, A50, EUCLID:53, EUCLID:54; ::_thesis: verum end; ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A35, A38, XREAL_1:64; then A51: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A36, A37, A38, XREAL_1:64; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A36, A37, A38, XREAL_1:64; then A52: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))))]| by A45, JGRAPH_2:3, JGRAPH_3:def_1; ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (- (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) >= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A36, A37, A38, A51, XREAL_1:24, XREAL_1:64; then A53: ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 >= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) >= - (- (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ) ) by XREAL_1:24; A54: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `1 by A36, A38, A40, XCMPLX_1:89; A55: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `2 by A37, A38, A40, XCMPLX_1:89; A56: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; not |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 by A36, A37, A38, A45, A51, XREAL_1:64; then A57: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2 > 0 by SQUARE_1:12; A58: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2 >= 0 by XREAL_1:63; (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) ^2) < 1 by A40, A41, A42, A43, A44, XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) < 1 by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) < 1 by A39, SQUARE_1:def_2; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) < 1 by A39, SQUARE_1:def_2; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) < 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A39, XREAL_1:68; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) < 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) < 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A39, XCMPLX_1:87; then A59: ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) < 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A39, XCMPLX_1:87; 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) = 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - 1 < (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2))) - 1 by A59, XREAL_1:9; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - 1) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) < (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) by A57, XREAL_1:68; then A60: (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) < (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2 by A57, XCMPLX_1:87; ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) = (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) ; then ( ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1 < 0 or ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) < 0 ) by A60, XREAL_1:49; then A61: (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) + 1 < 0 + 1 by A58, XREAL_1:6; then A62: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 < 1 ^2 by SQUARE_1:48; A63: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 > - (1 ^2) by A61, SQUARE_1:48; ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 < 1 & 1 > - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 ) ) by A53, A62, XXREAL_0:2; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 < 1 & - 1 < - (- (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2)) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 > - 1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) > - 1 ) ) by A63, XREAL_1:24, XXREAL_0:2; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 < 1 & - 1 < |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 > - 1 & - (- (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2)) < - (- 1) ) ) by XREAL_1:24; then |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| in Kb by A1, A62, A63; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A28, A52, A54, A55, A56, EUCLID:53; ::_thesis: verum end; caseA64: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) set px = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]|; A65: ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A64, JGRAPH_2:13; A66: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52; A67: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52; 1 + (((p2 `1) / (p2 `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; then A68: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by SQUARE_1:25; A69: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A70: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) = (p2 `1) / (p2 `2) by A66, A67, A68, XCMPLX_1:91; A71: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A68, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52 ; A72: p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A68, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52 ; A73: |.p2.| ^2 = ((p2 `2) ^2) + ((p2 `1) ^2) by JGRAPH_3:1; A74: |.p2.| ^2 < 1 ^2 by A29, SQUARE_1:16; A75: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_=_0_) assume that A76: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 and A77: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = 0 ; ::_thesis: contradiction A78: (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A76, EUCLID:52; (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A77, EUCLID:52; then p2 `1 = 0 by A68, XCMPLX_1:6; hence contradiction by A64, A68, A78, XCMPLX_1:6; ::_thesis: verum end; ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A65, A68, XREAL_1:64; then A79: ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A66, A67, A68, XREAL_1:64; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A66, A67, A68, XREAL_1:64; then A80: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))))]| by A75, JGRAPH_2:3, JGRAPH_3:4; ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (- (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) >= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A66, A67, A68, A79, XREAL_1:24, XREAL_1:64; then A81: ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 >= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) >= - (- (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ) ) by XREAL_1:24; A82: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `2 by A66, A68, A70, XCMPLX_1:89; A83: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `1 by A67, A68, A70, XCMPLX_1:89; A84: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; not |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 by A66, A67, A68, A75, A79, XREAL_1:64; then A85: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2 > 0 by SQUARE_1:12; A86: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 >= 0 by XREAL_1:63; (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) ^2) < 1 by A70, A71, A72, A73, A74, XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) < 1 by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) < 1 by A69, SQUARE_1:def_2; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) < 1 by A69, SQUARE_1:def_2; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) < 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A69, XREAL_1:68; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) < 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) < 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A69, XCMPLX_1:87; then A87: ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) < 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A69, XCMPLX_1:87; 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) = 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1 < (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2))) - 1 by A87, XREAL_1:9; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) < (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) by A85, XREAL_1:68; then A88: (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) < (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 by A85, XCMPLX_1:87; ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) = (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) ; then ( ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1 < 0 or ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) < 0 ) by A88, XREAL_1:49; then A89: (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) + 1 < 0 + 1 by A86, XREAL_1:6; then A90: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 < 1 ^2 by SQUARE_1:48; A91: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 > - (1 ^2) by A89, SQUARE_1:48; ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 < 1 & 1 > - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 ) ) by A81, A90, XXREAL_0:2; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 < 1 & - 1 < - (- (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1)) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 > - 1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) > - 1 ) ) by A91, XREAL_1:24, XXREAL_0:2; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 < 1 & - 1 < |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 > - 1 & - (- (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1)) < - (- 1) ) ) by XREAL_1:24; then |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| in Kb by A1, A90, A91; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A28, A80, A82, A83, A84, EUCLID:53; ::_thesis: verum end; end; end; hence y in Sq_Circ .: Kb by FUNCT_1:def_6; ::_thesis: verum end; theorem Th26: :: JGRAPH_6:26 for Kb, Cb being Subset of (TOP-REAL 2) st Kb = { p where p is Point of (TOP-REAL 2) : ( not - 1 <= p `1 or not p `1 <= 1 or not - 1 <= p `2 or not p `2 <= 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| > 1 } holds Sq_Circ .: Kb = Cb proof let Kb, Cb be Subset of (TOP-REAL 2); ::_thesis: ( Kb = { p where p is Point of (TOP-REAL 2) : ( not - 1 <= p `1 or not p `1 <= 1 or not - 1 <= p `2 or not p `2 <= 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| > 1 } implies Sq_Circ .: Kb = Cb ) assume A1: ( Kb = { p where p is Point of (TOP-REAL 2) : ( not - 1 <= p `1 or not p `1 <= 1 or not - 1 <= p `2 or not p `2 <= 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| > 1 } ) ; ::_thesis: Sq_Circ .: Kb = Cb thus Sq_Circ .: Kb c= Cb :: according to XBOOLE_0:def_10 ::_thesis: Cb c= Sq_Circ .: Kb proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Sq_Circ .: Kb or y in Cb ) assume y in Sq_Circ .: Kb ; ::_thesis: y in Cb then consider x being set such that x in dom Sq_Circ and A2: x in Kb and A3: y = Sq_Circ . x by FUNCT_1:def_6; consider q being Point of (TOP-REAL 2) such that A4: q = x and A5: ( not - 1 <= q `1 or not q `1 <= 1 or not - 1 <= q `2 or not q `2 <= 1 ) by A1, A2; now__::_thesis:_(_(_q_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_q_<>_0._(TOP-REAL_2)_&_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_>_1_)_)_or_(_q_<>_0._(TOP-REAL_2)_&_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_>_1_)_)_) percases ( q = 0. (TOP-REAL 2) or ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) or ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; case q = 0. (TOP-REAL 2) ; ::_thesis: contradiction hence contradiction by A5, EUCLID:52, EUCLID:54; ::_thesis: verum end; caseA6: ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| > 1 ) then A7: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by JGRAPH_3:def_1; A8: ( ( - 1 <= q `2 & q `2 <= 1 ) or - 1 > q `1 or q `1 > 1 ) proof assume A9: ( not - 1 <= q `2 or not q `2 <= 1 ) ; ::_thesis: ( - 1 > q `1 or q `1 > 1 ) now__::_thesis:_(_(_-_1_>_q_`2_&_(_-_1_>_q_`1_or_q_`1_>_1_)_)_or_(_q_`2_>_1_&_(_-_1_>_q_`1_or_q_`1_>_1_)_)_) percases ( - 1 > q `2 or q `2 > 1 ) by A9; caseA10: - 1 > q `2 ; ::_thesis: ( - 1 > q `1 or q `1 > 1 ) then ( - (q `1) < - 1 or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) by A6, XXREAL_0:2; hence ( - 1 > q `1 or q `1 > 1 ) by A10, XREAL_1:24, XXREAL_0:2; ::_thesis: verum end; case q `2 > 1 ; ::_thesis: ( - 1 > q `1 or q `1 > 1 ) then ( 1 < q `1 or 1 < - (q `1) ) by A6, XXREAL_0:2; then ( 1 < q `1 or - (- (q `1)) < - 1 ) by XREAL_1:24; hence ( - 1 > q `1 or q `1 > 1 ) ; ::_thesis: verum end; end; end; hence ( - 1 > q `1 or q `1 > 1 ) ; ::_thesis: verum end; A11: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52; A12: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52; A13: 1 + (((q `2) / (q `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A14: now__::_thesis:_not_q_`1_=_0 assume A15: q `1 = 0 ; ::_thesis: contradiction then q `2 = 0 by A6; hence contradiction by A6, A15, EUCLID:53, EUCLID:54; ::_thesis: verum end; then A16: (q `1) ^2 > 0 by SQUARE_1:12; (q `1) ^2 > 1 ^2 by A5, A8, SQUARE_1:47; then A17: sqrt ((q `1) ^2) > 1 by SQUARE_1:18, SQUARE_1:27; |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) by A11, A12, JGRAPH_3:1 .= (((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) by XCMPLX_1:76 .= (((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) by XCMPLX_1:76 .= (((q `1) ^2) / (1 + (((q `2) / (q `1)) ^2))) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) by A13, SQUARE_1:def_2 .= (((q `1) ^2) / (1 + (((q `2) / (q `1)) ^2))) + (((q `2) ^2) / (1 + (((q `2) / (q `1)) ^2))) by A13, SQUARE_1:def_2 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `2) / (q `1)) ^2)) by XCMPLX_1:62 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `2) ^2) / ((q `1) ^2))) by XCMPLX_1:76 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) / ((q `1) ^2)) + (((q `2) ^2) / ((q `1) ^2))) by A16, XCMPLX_1:60 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) + ((q `2) ^2)) / ((q `1) ^2)) by XCMPLX_1:62 .= ((q `1) ^2) * ((((q `1) ^2) + ((q `2) ^2)) / (((q `1) ^2) + ((q `2) ^2))) by XCMPLX_1:81 .= ((q `1) ^2) * 1 by A14, COMPLEX1:1, XCMPLX_1:60 .= (q `1) ^2 ; then |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| > 1 by A17, SQUARE_1:22; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| > 1 ) by A3, A4, A7; ::_thesis: verum end; caseA18: ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| > 1 ) then A19: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by JGRAPH_3:def_1; A20: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52; A21: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52; A22: 1 + (((q `1) / (q `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A23: q `2 <> 0 by A18; then A24: (q `2) ^2 > 0 by SQUARE_1:12; ( ( - 1 <= q `1 & q `1 <= 1 ) or - 1 > q `2 or q `2 > 1 ) proof assume A25: ( not - 1 <= q `1 or not q `1 <= 1 ) ; ::_thesis: ( - 1 > q `2 or q `2 > 1 ) now__::_thesis:_(_(_-_1_>_q_`1_&_(_-_1_>_q_`2_or_q_`2_>_1_)_)_or_(_q_`1_>_1_&_(_-_1_>_q_`2_or_q_`2_>_1_)_)_) percases ( - 1 > q `1 or q `1 > 1 ) by A25; caseA26: - 1 > q `1 ; ::_thesis: ( - 1 > q `2 or q `2 > 1 ) then ( q `2 < - 1 or ( q `1 < q `2 & - (q `2) < - (- (q `1)) ) ) by A18, XREAL_1:24, XXREAL_0:2; then ( - (q `2) < - 1 or - 1 > q `2 ) by A26, XXREAL_0:2; hence ( - 1 > q `2 or q `2 > 1 ) by XREAL_1:24; ::_thesis: verum end; caseA27: q `1 > 1 ; ::_thesis: ( - 1 > q `2 or q `2 > 1 ) ( ( - (- (q `1)) < - (q `2) & q `2 < q `1 ) or ( q `2 > q `1 & q `2 > - (q `1) ) ) by A18, XREAL_1:24; then ( 1 < - (q `2) or ( q `2 > q `1 & q `2 > - (q `1) ) ) by A27, XXREAL_0:2; then ( - 1 > - (- (q `2)) or 1 < q `2 ) by A27, XREAL_1:24, XXREAL_0:2; hence ( - 1 > q `2 or q `2 > 1 ) ; ::_thesis: verum end; end; end; hence ( - 1 > q `2 or q `2 > 1 ) ; ::_thesis: verum end; then (q `2) ^2 > 1 ^2 by A5, SQUARE_1:47; then A28: sqrt ((q `2) ^2) > 1 by SQUARE_1:18, SQUARE_1:27; |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) by A20, A21, JGRAPH_3:1 .= (((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) by XCMPLX_1:76 .= (((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) by XCMPLX_1:76 .= (((q `1) ^2) / (1 + (((q `1) / (q `2)) ^2))) + (((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) by A22, SQUARE_1:def_2 .= (((q `1) ^2) / (1 + (((q `1) / (q `2)) ^2))) + (((q `2) ^2) / (1 + (((q `1) / (q `2)) ^2))) by A22, SQUARE_1:def_2 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `1) / (q `2)) ^2)) by XCMPLX_1:62 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `1) ^2) / ((q `2) ^2))) by XCMPLX_1:76 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) / ((q `2) ^2)) + (((q `2) ^2) / ((q `2) ^2))) by A24, XCMPLX_1:60 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) + ((q `2) ^2)) / ((q `2) ^2)) by XCMPLX_1:62 .= ((q `2) ^2) * ((((q `1) ^2) + ((q `2) ^2)) / (((q `1) ^2) + ((q `2) ^2))) by XCMPLX_1:81 .= ((q `2) ^2) * 1 by A23, COMPLEX1:1, XCMPLX_1:60 .= (q `2) ^2 ; then |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| > 1 by A28, SQUARE_1:22; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| > 1 ) by A3, A4, A19; ::_thesis: verum end; end; end; hence y in Cb by A1; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Cb or y in Sq_Circ .: Kb ) assume y in Cb ; ::_thesis: y in Sq_Circ .: Kb then consider p2 being Point of (TOP-REAL 2) such that A29: p2 = y and A30: |.p2.| > 1 by A1; set q = p2; now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_) percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; case p2 = 0. (TOP-REAL 2) ; ::_thesis: contradiction hence contradiction by A30, TOPRNS_1:23; ::_thesis: verum end; caseA31: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) set px = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]|; A32: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52; A33: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52; 1 + (((p2 `2) / (p2 `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; then A34: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by SQUARE_1:25; A35: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A36: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) = (p2 `2) / (p2 `1) by A32, A33, A34, XCMPLX_1:91; A37: p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A34, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ; A38: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A34, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ; A39: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; A40: |.p2.| ^2 > 1 ^2 by A30, SQUARE_1:16; A41: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_=_0_) assume that A42: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 and A43: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = 0 ; ::_thesis: contradiction A44: (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A42, EUCLID:52; A45: (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A43, EUCLID:52; A46: p2 `1 = 0 by A34, A44, XCMPLX_1:6; p2 `2 = 0 by A34, A45, XCMPLX_1:6; hence contradiction by A31, A46, EUCLID:53, EUCLID:54; ::_thesis: verum end; ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A31, A34, XREAL_1:64; then A47: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A32, A33, A34, XREAL_1:64; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A32, A33, A34, XREAL_1:64; then A48: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))))]| by A41, JGRAPH_2:3, JGRAPH_3:def_1; A49: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `1 by A32, A34, A36, XCMPLX_1:89; A50: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `2 by A33, A34, A36, XCMPLX_1:89; A51: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; not |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 by A32, A33, A34, A41, A47, XREAL_1:64; then A52: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2 > 0 by SQUARE_1:12; A53: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2 >= 0 by XREAL_1:63; (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) ^2) > 1 by A36, A37, A38, A39, A40, XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) > 1 by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) > 1 by A35, SQUARE_1:def_2; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) > 1 by A35, SQUARE_1:def_2; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) > 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A35, XREAL_1:68; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) > 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) > 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A35, XCMPLX_1:87; then A54: ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) > 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A35, XCMPLX_1:87; 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) = 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - 1 > (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2))) - 1 by A54, XREAL_1:9; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - 1) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) > (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) by A52, XREAL_1:68; then A55: (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) > (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2 by A52, XCMPLX_1:87; ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) = (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) ; then ( ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1 > 0 or ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) < 0 ) by A55, XREAL_1:50; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) + 1 > 0 + 1 by A52, A53, XREAL_1:6; then ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 > 1 ^2 or |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 < - 1 ) by SQUARE_1:49; then |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| in Kb by A1; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A29, A48, A49, A50, A51, EUCLID:53; ::_thesis: verum end; caseA56: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) set px = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]|; A57: ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A56, JGRAPH_2:13; A58: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52; A59: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52; 1 + (((p2 `1) / (p2 `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; then A60: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by SQUARE_1:25; A61: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A62: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) = (p2 `1) / (p2 `2) by A58, A59, A60, XCMPLX_1:91; A63: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A60, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52 ; A64: p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A60, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52 ; A65: |.p2.| ^2 = ((p2 `2) ^2) + ((p2 `1) ^2) by JGRAPH_3:1; A66: |.p2.| ^2 > 1 ^2 by A30, SQUARE_1:16; A67: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_=_0_) assume that A68: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 and A69: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = 0 ; ::_thesis: contradiction A70: (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A68, EUCLID:52; (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A69, EUCLID:52; then p2 `1 = 0 by A60, XCMPLX_1:6; hence contradiction by A56, A60, A70, XCMPLX_1:6; ::_thesis: verum end; ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A57, A60, XREAL_1:64; then A71: ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A58, A59, A60, XREAL_1:64; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A58, A59, A60, XREAL_1:64; then A72: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))))]| by A67, JGRAPH_2:3, JGRAPH_3:4; A73: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `2 by A58, A60, A62, XCMPLX_1:89; A74: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `1 by A59, A60, A62, XCMPLX_1:89; A75: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; not |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 by A58, A59, A60, A67, A71, XREAL_1:64; then A76: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2 > 0 by SQUARE_1:12; A77: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 >= 0 by XREAL_1:63; (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) ^2) > 1 by A62, A63, A64, A65, A66, XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) > 1 by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) > 1 by A61, SQUARE_1:def_2; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) > 1 by A61, SQUARE_1:def_2; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) > 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A61, XREAL_1:68; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) > 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) > 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A61, XCMPLX_1:87; then A78: ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) > 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A61, XCMPLX_1:87; 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) = 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1 > (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2))) - 1 by A78, XREAL_1:9; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) > (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) by A76, XREAL_1:68; then A79: (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) > (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 by A76, XCMPLX_1:87; ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) = (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) ; then ( ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1 > 0 or ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) < 0 ) by A79, XREAL_1:50; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) + 1 > 0 + 1 by A76, A77, XREAL_1:6; then ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 > 1 ^2 or |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 < - 1 ) by SQUARE_1:49; then |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| in Kb by A1; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A29, A72, A73, A74, A75, EUCLID:53; ::_thesis: verum end; end; end; hence y in Sq_Circ .: Kb by FUNCT_1:def_6; ::_thesis: verum end; theorem Th27: :: JGRAPH_6:27 for Kb, Cb being Subset of (TOP-REAL 2) st Kb = { p where p is Point of (TOP-REAL 2) : ( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| <= 1 } holds Sq_Circ .: Kb = Cb proof let Kb, Cb be Subset of (TOP-REAL 2); ::_thesis: ( Kb = { p where p is Point of (TOP-REAL 2) : ( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| <= 1 } implies Sq_Circ .: Kb = Cb ) assume A1: ( Kb = { p where p is Point of (TOP-REAL 2) : ( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| <= 1 } ) ; ::_thesis: Sq_Circ .: Kb = Cb thus Sq_Circ .: Kb c= Cb :: according to XBOOLE_0:def_10 ::_thesis: Cb c= Sq_Circ .: Kb proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Sq_Circ .: Kb or y in Cb ) assume y in Sq_Circ .: Kb ; ::_thesis: y in Cb then consider x being set such that x in dom Sq_Circ and A2: x in Kb and A3: y = Sq_Circ . x by FUNCT_1:def_6; consider q being Point of (TOP-REAL 2) such that A4: q = x and A5: - 1 <= q `1 and A6: q `1 <= 1 and A7: - 1 <= q `2 and A8: q `2 <= 1 by A1, A2; now__::_thesis:_(_(_q_=_0._(TOP-REAL_2)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_<=_1_)_)_or_(_q_<>_0._(TOP-REAL_2)_&_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_<=_1_)_)_or_(_q_<>_0._(TOP-REAL_2)_&_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_<=_1_)_)_) percases ( q = 0. (TOP-REAL 2) or ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) or ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; caseA9: q = 0. (TOP-REAL 2) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| <= 1 ) then A10: Sq_Circ . q = q by JGRAPH_3:def_1; |.q.| = 0 by A9, TOPRNS_1:23; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| <= 1 ) by A3, A4, A10; ::_thesis: verum end; caseA11: ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| <= 1 ) then A12: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by JGRAPH_3:def_1; A13: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52; A14: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52; A15: 1 + (((q `2) / (q `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A16: now__::_thesis:_not_q_`1_=_0 assume A17: q `1 = 0 ; ::_thesis: contradiction then q `2 = 0 by A11; hence contradiction by A11, A17, EUCLID:53, EUCLID:54; ::_thesis: verum end; then A18: (q `1) ^2 > 0 by SQUARE_1:12; (q `1) ^2 <= 1 ^2 by A5, A6, SQUARE_1:49; then A19: sqrt ((q `1) ^2) <= 1 by A18, SQUARE_1:18, SQUARE_1:26; |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) by A13, A14, JGRAPH_3:1 .= (((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) by XCMPLX_1:76 .= (((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) by XCMPLX_1:76 .= (((q `1) ^2) / (1 + (((q `2) / (q `1)) ^2))) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) by A15, SQUARE_1:def_2 .= (((q `1) ^2) / (1 + (((q `2) / (q `1)) ^2))) + (((q `2) ^2) / (1 + (((q `2) / (q `1)) ^2))) by A15, SQUARE_1:def_2 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `2) / (q `1)) ^2)) by XCMPLX_1:62 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `2) ^2) / ((q `1) ^2))) by XCMPLX_1:76 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) / ((q `1) ^2)) + (((q `2) ^2) / ((q `1) ^2))) by A18, XCMPLX_1:60 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) + ((q `2) ^2)) / ((q `1) ^2)) by XCMPLX_1:62 .= ((q `1) ^2) * ((((q `1) ^2) + ((q `2) ^2)) / (((q `1) ^2) + ((q `2) ^2))) by XCMPLX_1:81 .= ((q `1) ^2) * 1 by A16, COMPLEX1:1, XCMPLX_1:60 .= (q `1) ^2 ; then |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| <= 1 by A19, SQUARE_1:22; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| <= 1 ) by A3, A4, A12; ::_thesis: verum end; caseA20: ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| <= 1 ) then A21: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by JGRAPH_3:def_1; A22: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52; A23: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52; A24: 1 + (((q `1) / (q `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A25: q `2 <> 0 by A20; then A26: (q `2) ^2 > 0 by SQUARE_1:12; (q `2) ^2 <= 1 ^2 by A7, A8, SQUARE_1:49; then A27: sqrt ((q `2) ^2) <= 1 by A26, SQUARE_1:18, SQUARE_1:26; |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) by A22, A23, JGRAPH_3:1 .= (((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) by XCMPLX_1:76 .= (((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) by XCMPLX_1:76 .= (((q `1) ^2) / (1 + (((q `1) / (q `2)) ^2))) + (((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) by A24, SQUARE_1:def_2 .= (((q `1) ^2) / (1 + (((q `1) / (q `2)) ^2))) + (((q `2) ^2) / (1 + (((q `1) / (q `2)) ^2))) by A24, SQUARE_1:def_2 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `1) / (q `2)) ^2)) by XCMPLX_1:62 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `1) ^2) / ((q `2) ^2))) by XCMPLX_1:76 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) / ((q `2) ^2)) + (((q `2) ^2) / ((q `2) ^2))) by A26, XCMPLX_1:60 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) + ((q `2) ^2)) / ((q `2) ^2)) by XCMPLX_1:62 .= ((q `2) ^2) * ((((q `1) ^2) + ((q `2) ^2)) / (((q `1) ^2) + ((q `2) ^2))) by XCMPLX_1:81 .= ((q `2) ^2) * 1 by A25, COMPLEX1:1, XCMPLX_1:60 .= (q `2) ^2 ; then |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| <= 1 by A27, SQUARE_1:22; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| <= 1 ) by A3, A4, A21; ::_thesis: verum end; end; end; hence y in Cb by A1; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Cb or y in Sq_Circ .: Kb ) assume y in Cb ; ::_thesis: y in Sq_Circ .: Kb then consider p2 being Point of (TOP-REAL 2) such that A28: p2 = y and A29: |.p2.| <= 1 by A1; set q = p2; now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_) percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; caseA30: p2 = 0. (TOP-REAL 2) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) then A31: p2 `1 = 0 by EUCLID:52, EUCLID:54; p2 `2 = 0 by A30, EUCLID:52, EUCLID:54; then A32: y in Kb by A1, A28, A31; A33: (Sq_Circ ") . y = y by A28, A30, JGRAPH_3:28; A34: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; y = Sq_Circ . y by A28, A33, FUNCT_1:35, JGRAPH_3:43; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A32, A34; ::_thesis: verum end; caseA35: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) set px = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]|; A36: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52; A37: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52; 1 + (((p2 `2) / (p2 `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; then A38: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by SQUARE_1:25; A39: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A40: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) = (p2 `2) / (p2 `1) by A36, A37, A38, XCMPLX_1:91; A41: p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A38, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ; A42: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A38, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ; A43: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; A44: |.p2.| ^2 <= 1 ^2 by A29, SQUARE_1:15; A45: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_=_0_) assume that A46: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 and A47: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = 0 ; ::_thesis: contradiction A48: (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A46, EUCLID:52; A49: (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A47, EUCLID:52; A50: p2 `1 = 0 by A38, A48, XCMPLX_1:6; p2 `2 = 0 by A38, A49, XCMPLX_1:6; hence contradiction by A35, A50, EUCLID:53, EUCLID:54; ::_thesis: verum end; ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A35, A38, XREAL_1:64; then A51: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A36, A37, A38, XREAL_1:64; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A36, A37, A38, XREAL_1:64; then A52: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))))]| by A45, JGRAPH_2:3, JGRAPH_3:def_1; ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (- (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) >= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A36, A37, A38, A51, XREAL_1:24, XREAL_1:64; then A53: ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 >= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) >= - (- (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ) ) by XREAL_1:24; A54: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `1 by A36, A38, A40, XCMPLX_1:89; A55: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `2 by A37, A38, A40, XCMPLX_1:89; A56: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; not |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 by A36, A37, A38, A45, A51, XREAL_1:64; then A57: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2 > 0 by SQUARE_1:12; A58: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2 >= 0 by XREAL_1:63; (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) ^2) <= 1 by A40, A41, A42, A43, A44, XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) <= 1 by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) <= 1 by A39, SQUARE_1:def_2; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) <= 1 by A39, SQUARE_1:def_2; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) <= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A39, XREAL_1:64; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) <= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) <= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A39, XCMPLX_1:87; then A59: ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) <= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A39, XCMPLX_1:87; 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) = 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - 1 <= (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2))) - 1 by A59, XREAL_1:9; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - 1) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) <= (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) by A57, XREAL_1:64; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) <= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2 by A57, XCMPLX_1:87; then A60: ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) - 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) <= 0 by XREAL_1:47; ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) = (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) ; then ( ( ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1 <= 0 & ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1 >= 0 ) or ( ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1 <= 0 & ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) >= 0 ) or ( ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) <= 0 & ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1 >= 0 ) or ( ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) <= 0 & ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) >= 0 ) ) by A60, XREAL_1:129, XREAL_1:130; then A61: (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) + 1 <= 0 + 1 by A58, XREAL_1:7; then A62: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ^2 by SQUARE_1:47; A63: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 >= - (1 ^2) by A61, SQUARE_1:47; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 & 1 >= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= - 1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 ) ) by A53, A62, XXREAL_0:2; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 & - 1 <= - (- (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2)) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= - 1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) >= - 1 ) ) by A63, XREAL_1:24, XXREAL_0:2; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= - 1 & - (- (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2)) <= - (- 1) ) ) by XREAL_1:24; then |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| in Kb by A1, A62, A63; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A28, A52, A54, A55, A56, EUCLID:53; ::_thesis: verum end; caseA64: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) set px = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]|; A65: ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A64, JGRAPH_2:13; A66: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52; A67: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52; 1 + (((p2 `1) / (p2 `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; then A68: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by SQUARE_1:25; A69: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A70: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) = (p2 `1) / (p2 `2) by A66, A67, A68, XCMPLX_1:91; A71: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A68, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52 ; A72: p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A68, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52 ; A73: |.p2.| ^2 = ((p2 `2) ^2) + ((p2 `1) ^2) by JGRAPH_3:1; A74: |.p2.| ^2 <= 1 ^2 by A29, SQUARE_1:15; A75: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_=_0_) assume that A76: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 and A77: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = 0 ; ::_thesis: contradiction A78: (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A76, EUCLID:52; (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A77, EUCLID:52; then p2 `1 = 0 by A68, XCMPLX_1:6; hence contradiction by A64, A68, A78, XCMPLX_1:6; ::_thesis: verum end; ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A65, A68, XREAL_1:64; then A79: ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A66, A67, A68, XREAL_1:64; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A66, A67, A68, XREAL_1:64; then A80: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))))]| by A75, JGRAPH_2:3, JGRAPH_3:4; ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (- (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) >= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A66, A67, A68, A79, XREAL_1:24, XREAL_1:64; then A81: ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 >= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) >= - (- (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ) ) by XREAL_1:24; A82: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `2 by A66, A68, A70, XCMPLX_1:89; A83: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `1 by A67, A68, A70, XCMPLX_1:89; A84: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; not |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 by A66, A67, A68, A75, A79, XREAL_1:64; then A85: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2 > 0 by SQUARE_1:12; A86: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 >= 0 by XREAL_1:63; (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) ^2) <= 1 by A70, A71, A72, A73, A74, XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) <= 1 by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) <= 1 by A69, SQUARE_1:def_2; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) <= 1 by A69, SQUARE_1:def_2; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) <= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A69, XREAL_1:64; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) <= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) <= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A69, XCMPLX_1:87; then A87: ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) <= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A69, XCMPLX_1:87; 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) = 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1 <= (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2))) - 1 by A87, XREAL_1:9; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) <= (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) by A85, XREAL_1:64; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) <= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 by A85, XCMPLX_1:87; then A88: ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) - 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) <= 0 by XREAL_1:47; ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) = (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) ; then ( ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1 <= 0 or ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) <= 0 ) by A88, XREAL_1:129; then A89: (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) + 1 <= 0 + 1 by A86, XREAL_1:7; then A90: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ^2 by SQUARE_1:47; A91: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 >= - (1 ^2) by A89, SQUARE_1:47; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 & 1 >= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= - 1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 ) ) by A81, A90, XXREAL_0:2; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 & - 1 <= - (- (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1)) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= - 1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) >= - 1 ) ) by A91, XREAL_1:24, XXREAL_0:2; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= - 1 & - (- (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1)) <= - (- 1) ) ) by XREAL_1:24; then |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| in Kb by A1, A90, A91; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A28, A80, A82, A83, A84, EUCLID:53; ::_thesis: verum end; end; end; hence y in Sq_Circ .: Kb by FUNCT_1:def_6; ::_thesis: verum end; theorem Th28: :: JGRAPH_6:28 for Kb, Cb being Subset of (TOP-REAL 2) st Kb = { p where p is Point of (TOP-REAL 2) : ( not - 1 < p `1 or not p `1 < 1 or not - 1 < p `2 or not p `2 < 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| >= 1 } holds Sq_Circ .: Kb = Cb proof let Kb, Cb be Subset of (TOP-REAL 2); ::_thesis: ( Kb = { p where p is Point of (TOP-REAL 2) : ( not - 1 < p `1 or not p `1 < 1 or not - 1 < p `2 or not p `2 < 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| >= 1 } implies Sq_Circ .: Kb = Cb ) assume A1: ( Kb = { p where p is Point of (TOP-REAL 2) : ( not - 1 < p `1 or not p `1 < 1 or not - 1 < p `2 or not p `2 < 1 ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| >= 1 } ) ; ::_thesis: Sq_Circ .: Kb = Cb thus Sq_Circ .: Kb c= Cb :: according to XBOOLE_0:def_10 ::_thesis: Cb c= Sq_Circ .: Kb proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Sq_Circ .: Kb or y in Cb ) assume y in Sq_Circ .: Kb ; ::_thesis: y in Cb then consider x being set such that x in dom Sq_Circ and A2: x in Kb and A3: y = Sq_Circ . x by FUNCT_1:def_6; consider q being Point of (TOP-REAL 2) such that A4: q = x and A5: ( not - 1 < q `1 or not q `1 < 1 or not - 1 < q `2 or not q `2 < 1 ) by A1, A2; now__::_thesis:_(_(_q_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_q_<>_0._(TOP-REAL_2)_&_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_>=_1_)_)_or_(_q_<>_0._(TOP-REAL_2)_&_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_ (_p2_=_y_&_|.p2.|_>=_1_)_)_) percases ( q = 0. (TOP-REAL 2) or ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) or ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; caseA6: q = 0. (TOP-REAL 2) ; ::_thesis: contradiction then A7: q `1 = 0 by EUCLID:52, EUCLID:54; q `2 = 0 by A6, EUCLID:52, EUCLID:54; hence contradiction by A5, A7; ::_thesis: verum end; caseA8: ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| >= 1 ) then A9: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by JGRAPH_3:def_1; A10: ( ( - 1 < q `2 & q `2 < 1 ) or - 1 >= q `1 or q `1 >= 1 ) proof assume A11: ( not - 1 < q `2 or not q `2 < 1 ) ; ::_thesis: ( - 1 >= q `1 or q `1 >= 1 ) now__::_thesis:_(_(_-_1_>=_q_`2_&_(_-_1_>=_q_`1_or_q_`1_>=_1_)_)_or_(_q_`2_>=_1_&_(_-_1_>=_q_`1_or_q_`1_>=_1_)_)_) percases ( - 1 >= q `2 or q `2 >= 1 ) by A11; caseA12: - 1 >= q `2 ; ::_thesis: ( - 1 >= q `1 or q `1 >= 1 ) then ( - (q `1) <= - 1 or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) by A8, XXREAL_0:2; hence ( - 1 >= q `1 or q `1 >= 1 ) by A12, XREAL_1:24, XXREAL_0:2; ::_thesis: verum end; case q `2 >= 1 ; ::_thesis: ( - 1 >= q `1 or q `1 >= 1 ) then ( 1 <= q `1 or 1 <= - (q `1) ) by A8, XXREAL_0:2; then ( 1 <= q `1 or - (- (q `1)) <= - 1 ) by XREAL_1:24; hence ( - 1 >= q `1 or q `1 >= 1 ) ; ::_thesis: verum end; end; end; hence ( - 1 >= q `1 or q `1 >= 1 ) ; ::_thesis: verum end; A13: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52; A14: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52; A15: 1 + (((q `2) / (q `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A16: now__::_thesis:_not_q_`1_=_0 assume A17: q `1 = 0 ; ::_thesis: contradiction then q `2 = 0 by A8; hence contradiction by A8, A17, EUCLID:53, EUCLID:54; ::_thesis: verum end; then A18: (q `1) ^2 > 0 by SQUARE_1:12; (q `1) ^2 >= 1 ^2 by A5, A10, SQUARE_1:48; then A19: sqrt ((q `1) ^2) >= 1 by SQUARE_1:18, SQUARE_1:26; |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) by A13, A14, JGRAPH_3:1 .= (((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2) by XCMPLX_1:76 .= (((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) by XCMPLX_1:76 .= (((q `1) ^2) / (1 + (((q `2) / (q `1)) ^2))) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2)) by A15, SQUARE_1:def_2 .= (((q `1) ^2) / (1 + (((q `2) / (q `1)) ^2))) + (((q `2) ^2) / (1 + (((q `2) / (q `1)) ^2))) by A15, SQUARE_1:def_2 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `2) / (q `1)) ^2)) by XCMPLX_1:62 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `2) ^2) / ((q `1) ^2))) by XCMPLX_1:76 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) / ((q `1) ^2)) + (((q `2) ^2) / ((q `1) ^2))) by A18, XCMPLX_1:60 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) + ((q `2) ^2)) / ((q `1) ^2)) by XCMPLX_1:62 .= ((q `1) ^2) * ((((q `1) ^2) + ((q `2) ^2)) / (((q `1) ^2) + ((q `2) ^2))) by XCMPLX_1:81 .= ((q `1) ^2) * 1 by A16, COMPLEX1:1, XCMPLX_1:60 .= (q `1) ^2 ; then |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| >= 1 by A19, SQUARE_1:22; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| >= 1 ) by A3, A4, A9; ::_thesis: verum end; caseA20: ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| >= 1 ) then A21: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by JGRAPH_3:def_1; A22: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52; A23: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52; A24: 1 + (((q `1) / (q `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A25: q `2 <> 0 by A20; then A26: (q `2) ^2 > 0 by SQUARE_1:12; ( ( - 1 < q `1 & q `1 < 1 ) or - 1 >= q `2 or q `2 >= 1 ) proof assume A27: ( not - 1 < q `1 or not q `1 < 1 ) ; ::_thesis: ( - 1 >= q `2 or q `2 >= 1 ) now__::_thesis:_(_(_-_1_>=_q_`1_&_(_-_1_>=_q_`2_or_q_`2_>=_1_)_)_or_(_q_`1_>=_1_&_(_-_1_>=_q_`2_or_q_`2_>=_1_)_)_) percases ( - 1 >= q `1 or q `1 >= 1 ) by A27; caseA28: - 1 >= q `1 ; ::_thesis: ( - 1 >= q `2 or q `2 >= 1 ) then ( q `2 <= - 1 or ( q `1 < q `2 & - (q `2) <= - (- (q `1)) ) ) by A20, XREAL_1:24, XXREAL_0:2; then ( - (q `2) <= - 1 or - 1 >= q `2 ) by A28, XXREAL_0:2; hence ( - 1 >= q `2 or q `2 >= 1 ) by XREAL_1:24; ::_thesis: verum end; caseA29: q `1 >= 1 ; ::_thesis: ( - 1 >= q `2 or q `2 >= 1 ) ( ( - (- (q `1)) <= - (q `2) & q `2 <= q `1 ) or ( q `2 >= q `1 & q `2 >= - (q `1) ) ) by A20, XREAL_1:24; then ( 1 <= - (q `2) or ( q `2 >= q `1 & q `2 >= - (q `1) ) ) by A29, XXREAL_0:2; then ( - 1 >= - (- (q `2)) or 1 <= q `2 ) by A29, XREAL_1:24, XXREAL_0:2; hence ( - 1 >= q `2 or q `2 >= 1 ) ; ::_thesis: verum end; end; end; hence ( - 1 >= q `2 or q `2 >= 1 ) ; ::_thesis: verum end; then (q `2) ^2 >= 1 ^2 by A5, SQUARE_1:48; then A30: sqrt ((q `2) ^2) >= 1 by SQUARE_1:18, SQUARE_1:26; |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) by A22, A23, JGRAPH_3:1 .= (((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2) by XCMPLX_1:76 .= (((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) by XCMPLX_1:76 .= (((q `1) ^2) / (1 + (((q `1) / (q `2)) ^2))) + (((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2)) by A24, SQUARE_1:def_2 .= (((q `1) ^2) / (1 + (((q `1) / (q `2)) ^2))) + (((q `2) ^2) / (1 + (((q `1) / (q `2)) ^2))) by A24, SQUARE_1:def_2 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `1) / (q `2)) ^2)) by XCMPLX_1:62 .= (((q `1) ^2) + ((q `2) ^2)) / (1 + (((q `1) ^2) / ((q `2) ^2))) by XCMPLX_1:76 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) / ((q `2) ^2)) + (((q `2) ^2) / ((q `2) ^2))) by A26, XCMPLX_1:60 .= (((q `1) ^2) + ((q `2) ^2)) / ((((q `1) ^2) + ((q `2) ^2)) / ((q `2) ^2)) by XCMPLX_1:62 .= ((q `2) ^2) * ((((q `1) ^2) + ((q `2) ^2)) / (((q `1) ^2) + ((q `2) ^2))) by XCMPLX_1:81 .= ((q `2) ^2) * 1 by A25, COMPLEX1:1, XCMPLX_1:60 .= (q `2) ^2 ; then |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| >= 1 by A30, SQUARE_1:22; hence ex p2 being Point of (TOP-REAL 2) st ( p2 = y & |.p2.| >= 1 ) by A3, A4, A21; ::_thesis: verum end; end; end; hence y in Cb by A1; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Cb or y in Sq_Circ .: Kb ) assume y in Cb ; ::_thesis: y in Sq_Circ .: Kb then consider p2 being Point of (TOP-REAL 2) such that A31: p2 = y and A32: |.p2.| >= 1 by A1; set q = p2; now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_ex_x_being_set_st_ (_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_) percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; case p2 = 0. (TOP-REAL 2) ; ::_thesis: contradiction hence contradiction by A32, TOPRNS_1:23; ::_thesis: verum end; caseA33: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) set px = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]|; A34: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52; A35: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52; 1 + (((p2 `2) / (p2 `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; then A36: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by SQUARE_1:25; A37: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A38: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) = (p2 `2) / (p2 `1) by A34, A35, A36, XCMPLX_1:91; A39: p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A36, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ; A40: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A36, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ; A41: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; A42: |.p2.| ^2 >= 1 ^2 by A32, SQUARE_1:15; A43: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_=_0_) assume that A44: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 and A45: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = 0 ; ::_thesis: contradiction A46: (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A44, EUCLID:52; A47: (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A45, EUCLID:52; A48: p2 `1 = 0 by A36, A46, XCMPLX_1:6; p2 `2 = 0 by A36, A47, XCMPLX_1:6; hence contradiction by A33, A48, EUCLID:53, EUCLID:54; ::_thesis: verum end; ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A33, A36, XREAL_1:64; then A49: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A34, A35, A36, XREAL_1:64; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A34, A35, A36, XREAL_1:64; then A50: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))))]| by A43, JGRAPH_2:3, JGRAPH_3:def_1; A51: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `1 by A34, A36, A38, XCMPLX_1:89; A52: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `2 by A35, A36, A38, XCMPLX_1:89; A53: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; not |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 by A34, A35, A36, A43, A49, XREAL_1:64; then A54: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2 > 0 by SQUARE_1:12; then A55: ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) > 0 by XREAL_1:34, XREAL_1:63; (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) ^2) >= 1 by A38, A39, A40, A41, A42, XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) >= 1 by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) >= 1 by A37, SQUARE_1:def_2; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) >= 1 by A37, SQUARE_1:def_2; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) >= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A37, XREAL_1:64; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) >= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) >= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A37, XCMPLX_1:87; then A56: ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) >= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A37, XCMPLX_1:87; 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) = 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - 1 >= (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2))) - 1 by A56, XREAL_1:9; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - 1) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) >= (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) by A54, XREAL_1:64; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) >= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2 by A54, XCMPLX_1:87; then A57: ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) - 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) >= 0 by XREAL_1:48; ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) - (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) * 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) = (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1 >= 0 by A55, A57, XREAL_1:132; then (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) + 1 >= 0 + 1 by XREAL_1:7; then ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 >= 1 ^2 or |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= - 1 ) by SQUARE_1:50; then |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| in Kb by A1; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A31, A50, A51, A52, A53, EUCLID:53; ::_thesis: verum end; caseA58: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) set px = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]|; A59: ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A58, JGRAPH_2:13; A60: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52; A61: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52; 1 + (((p2 `1) / (p2 `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; then A62: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by SQUARE_1:25; A63: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2) > 0 by XREAL_1:34, XREAL_1:63; A64: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) = (p2 `1) / (p2 `2) by A60, A61, A62, XCMPLX_1:91; A65: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A62, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52 ; A66: p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A62, XCMPLX_1:89 .= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52 ; A67: |.p2.| ^2 = ((p2 `2) ^2) + ((p2 `1) ^2) by JGRAPH_3:1; A68: |.p2.| ^2 >= 1 ^2 by A32, SQUARE_1:15; A69: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_=_0_) assume that A70: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 and A71: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = 0 ; ::_thesis: contradiction A72: (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A70, EUCLID:52; (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A71, EUCLID:52; then p2 `1 = 0 by A62, XCMPLX_1:6; hence contradiction by A58, A62, A72, XCMPLX_1:6; ::_thesis: verum end; ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A59, A62, XREAL_1:64; then A73: ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A60, A61, A62, XREAL_1:64; then ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A60, A61, A62, XREAL_1:64; then A74: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))))]| by A69, JGRAPH_2:3, JGRAPH_3:4; A75: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `2 by A60, A62, A64, XCMPLX_1:89; A76: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `1 by A61, A62, A64, XCMPLX_1:89; A77: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; not |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 by A60, A61, A62, A69, A73, XREAL_1:64; then A78: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2 > 0 by SQUARE_1:12; A79: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 >= 0 by XREAL_1:63; (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) ^2) >= 1 by A64, A65, A66, A67, A68, XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) >= 1 by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) >= 1 by A63, SQUARE_1:def_2; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) >= 1 by A63, SQUARE_1:def_2; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) >= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A63, XREAL_1:64; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) >= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) ; then ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) >= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A63, XCMPLX_1:87; then A80: ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) >= 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A63, XCMPLX_1:87; 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) = 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) by XCMPLX_1:76; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1 >= (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2))) - 1 by A80, XREAL_1:9; then ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) >= (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) by A78, XREAL_1:64; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) >= (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 by A78, XCMPLX_1:87; then A81: ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) - 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) >= 0 by XREAL_1:48; ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2)) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) * 1))) - ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) = (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) ; then ( ( ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1 >= 0 & ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) >= 0 ) or ( ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1 <= 0 & ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) <= 0 ) ) by A81, XREAL_1:132; then (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) + 1 >= 0 + 1 by A78, A79, XREAL_1:7; then ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 >= 1 ^2 or |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= - 1 ) by SQUARE_1:50; then |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| in Kb by A1; hence ex x being set st ( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A31, A74, A75, A76, A77, EUCLID:53; ::_thesis: verum end; end; end; hence y in Sq_Circ .: Kb by FUNCT_1:def_6; ::_thesis: verum end; theorem :: JGRAPH_6:29 for P0, P1, P01, P11, K0, K1, K01, K11 being Subset of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & P0 = inside_of_circle (0,0,1) & P1 = outside_of_circle (0,0,1) & P01 = closed_inside_of_circle (0,0,1) & P11 = closed_outside_of_circle (0,0,1) & K0 = inside_of_rectangle ((- 1),1,(- 1),1) & K1 = outside_of_rectangle ((- 1),1,(- 1),1) & K01 = closed_inside_of_rectangle ((- 1),1,(- 1),1) & K11 = closed_outside_of_rectangle ((- 1),1,(- 1),1) & f = Sq_Circ holds ( f .: (rectangle ((- 1),1,(- 1),1)) = P & (f ") .: P = rectangle ((- 1),1,(- 1),1) & f .: K0 = P0 & (f ") .: P0 = K0 & f .: K1 = P1 & (f ") .: P1 = K1 & f .: K01 = P01 & f .: K11 = P11 & (f ") .: P01 = K01 & (f ") .: P11 = K11 ) proof let P0, P1, P01, P11, K0, K1, K01, K11 be Subset of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & P0 = inside_of_circle (0,0,1) & P1 = outside_of_circle (0,0,1) & P01 = closed_inside_of_circle (0,0,1) & P11 = closed_outside_of_circle (0,0,1) & K0 = inside_of_rectangle ((- 1),1,(- 1),1) & K1 = outside_of_rectangle ((- 1),1,(- 1),1) & K01 = closed_inside_of_rectangle ((- 1),1,(- 1),1) & K11 = closed_outside_of_rectangle ((- 1),1,(- 1),1) & f = Sq_Circ holds ( f .: (rectangle ((- 1),1,(- 1),1)) = P & (f ") .: P = rectangle ((- 1),1,(- 1),1) & f .: K0 = P0 & (f ") .: P0 = K0 & f .: K1 = P1 & (f ") .: P1 = K1 & f .: K01 = P01 & f .: K11 = P11 & (f ") .: P01 = K01 & (f ") .: P11 = K11 ) let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & P0 = inside_of_circle (0,0,1) & P1 = outside_of_circle (0,0,1) & P01 = closed_inside_of_circle (0,0,1) & P11 = closed_outside_of_circle (0,0,1) & K0 = inside_of_rectangle ((- 1),1,(- 1),1) & K1 = outside_of_rectangle ((- 1),1,(- 1),1) & K01 = closed_inside_of_rectangle ((- 1),1,(- 1),1) & K11 = closed_outside_of_rectangle ((- 1),1,(- 1),1) & f = Sq_Circ holds ( f .: (rectangle ((- 1),1,(- 1),1)) = P & (f ") .: P = rectangle ((- 1),1,(- 1),1) & f .: K0 = P0 & (f ") .: P0 = K0 & f .: K1 = P1 & (f ") .: P1 = K1 & f .: K01 = P01 & f .: K11 = P11 & (f ") .: P01 = K01 & (f ") .: P11 = K11 ) let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( P = circle (0,0,1) & P0 = inside_of_circle (0,0,1) & P1 = outside_of_circle (0,0,1) & P01 = closed_inside_of_circle (0,0,1) & P11 = closed_outside_of_circle (0,0,1) & K0 = inside_of_rectangle ((- 1),1,(- 1),1) & K1 = outside_of_rectangle ((- 1),1,(- 1),1) & K01 = closed_inside_of_rectangle ((- 1),1,(- 1),1) & K11 = closed_outside_of_rectangle ((- 1),1,(- 1),1) & f = Sq_Circ implies ( f .: (rectangle ((- 1),1,(- 1),1)) = P & (f ") .: P = rectangle ((- 1),1,(- 1),1) & f .: K0 = P0 & (f ") .: P0 = K0 & f .: K1 = P1 & (f ") .: P1 = K1 & f .: K01 = P01 & f .: K11 = P11 & (f ") .: P01 = K01 & (f ") .: P11 = K11 ) ) assume that A1: P = circle (0,0,1) and A2: P0 = inside_of_circle (0,0,1) and A3: P1 = outside_of_circle (0,0,1) and A4: P01 = closed_inside_of_circle (0,0,1) and A5: P11 = closed_outside_of_circle (0,0,1) and A6: K0 = inside_of_rectangle ((- 1),1,(- 1),1) and A7: K1 = outside_of_rectangle ((- 1),1,(- 1),1) and A8: K01 = closed_inside_of_rectangle ((- 1),1,(- 1),1) and A9: K11 = closed_outside_of_rectangle ((- 1),1,(- 1),1) and A10: f = Sq_Circ ; ::_thesis: ( f .: (rectangle ((- 1),1,(- 1),1)) = P & (f ") .: P = rectangle ((- 1),1,(- 1),1) & f .: K0 = P0 & (f ") .: P0 = K0 & f .: K1 = P1 & (f ") .: P1 = K1 & f .: K01 = P01 & f .: K11 = P11 & (f ") .: P01 = K01 & (f ") .: P11 = K11 ) set K = rectangle ((- 1),1,(- 1),1); A11: P0 = { p where p is Point of (TOP-REAL 2) : |.p.| < 1 } by A2, Th24; A12: P01 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } by A4, Th24; A13: P1 = { p where p is Point of (TOP-REAL 2) : |.p.| > 1 } by A3, Th24; A14: P11 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } by A5, Th24; defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `1 = - 1 & $1 `2 <= 1 & $1 `2 >= - 1 ) or ( $1 `1 <= 1 & $1 `1 >= - 1 & $1 `2 = 1 ) or ( $1 `1 <= 1 & $1 `1 >= - 1 & $1 `2 = - 1 ) or ( $1 `1 = 1 & $1 `2 <= 1 & $1 `2 >= - 1 ) ); defpred S2[ Point of (TOP-REAL 2)] means ( ( - 1 = $1 `1 & - 1 <= $1 `2 & $1 `2 <= 1 ) or ( $1 `1 = 1 & - 1 <= $1 `2 & $1 `2 <= 1 ) or ( - 1 = $1 `2 & - 1 <= $1 `1 & $1 `1 <= 1 ) or ( 1 = $1 `2 & - 1 <= $1 `1 & $1 `1 <= 1 ) ); deffunc H1( set ) -> set = $1; A15: for p being Element of (TOP-REAL 2) holds ( S1[p] iff S2[p] ) ; A16: rectangle ((- 1),1,(- 1),1) = { H1(p) where p is Point of (TOP-REAL 2) : S1[p] } by SPPOL_2:54 .= { H1(q) where q is Point of (TOP-REAL 2) : S2[q] } from FRAENKEL:sch_3(A15) ; defpred S3[ Point of (TOP-REAL 2)] means |.$1.| = 1; defpred S4[ Point of (TOP-REAL 2)] means |.($1 - |[0,0]|).| = 1; A17: for p being Point of (TOP-REAL 2) holds ( S4[p] iff S3[p] ) by EUCLID:54, RLVECT_1:13; P = { H1(p) where p is Point of (TOP-REAL 2) : S4[p] } by A1 .= { H1(p2) where p2 is Point of (TOP-REAL 2) : S3[p2] } from FRAENKEL:sch_3(A17) ; then A18: f .: (rectangle ((- 1),1,(- 1),1)) = P by A10, A16, JGRAPH_3:23; A19: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A20: f .: K0 = P0 by A6, A10, A11, Th25; f .: K1 = P1 by A7, A10, A13, Th26; hence ( f .: (rectangle ((- 1),1,(- 1),1)) = P & (f ") .: P = rectangle ((- 1),1,(- 1),1) & f .: K0 = P0 & (f ") .: P0 = K0 & f .: K1 = P1 & (f ") .: P1 = K1 ) by A10, A18, A19, A20, FUNCT_1:107; ::_thesis: ( f .: K01 = P01 & f .: K11 = P11 & (f ") .: P01 = K01 & (f ") .: P11 = K11 ) A21: f .: K01 = P01 by A8, A10, A12, Th27; f .: K11 = P11 by A9, A10, A14, Th28; hence ( f .: K01 = P01 & f .: K11 = P11 & (f ") .: P01 = K01 & (f ") .: P11 = K11 ) by A10, A19, A21, FUNCT_1:107; ::_thesis: verum end; begin theorem Th30: :: JGRAPH_6:30 for a, b, c, d being real number st a <= b & c <= d holds ( LSeg (|[a,c]|,|[a,d]|) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = a & p1 `2 <= d & p1 `2 >= c ) } & LSeg (|[a,d]|,|[b,d]|) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) } & LSeg (|[a,c]|,|[b,c]|) = { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 <= b & q1 `1 >= a & q1 `2 = c ) } & LSeg (|[b,c]|,|[b,d]|) = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = b & q2 `2 <= d & q2 `2 >= c ) } ) proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies ( LSeg (|[a,c]|,|[a,d]|) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = a & p1 `2 <= d & p1 `2 >= c ) } & LSeg (|[a,d]|,|[b,d]|) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) } & LSeg (|[a,c]|,|[b,c]|) = { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 <= b & q1 `1 >= a & q1 `2 = c ) } & LSeg (|[b,c]|,|[b,d]|) = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = b & q2 `2 <= d & q2 `2 >= c ) } ) ) assume that A1: a <= b and A2: c <= d ; ::_thesis: ( LSeg (|[a,c]|,|[a,d]|) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = a & p1 `2 <= d & p1 `2 >= c ) } & LSeg (|[a,d]|,|[b,d]|) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) } & LSeg (|[a,c]|,|[b,c]|) = { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 <= b & q1 `1 >= a & q1 `2 = c ) } & LSeg (|[b,c]|,|[b,d]|) = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = b & q2 `2 <= d & q2 `2 >= c ) } ) set L1 = { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } ; set L2 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } ; set L3 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } ; set L4 = { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } ; set p0 = |[a,c]|; set p01 = |[a,d]|; set p10 = |[b,c]|; set p1 = |[b,d]|; A3: |[a,d]| `1 = a by EUCLID:52; A4: |[a,d]| `2 = d by EUCLID:52; A5: |[b,c]| `1 = b by EUCLID:52; A6: |[b,c]| `2 = c by EUCLID:52; A7: { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } c= LSeg (|[a,c]|,|[a,d]|) proof let a2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } or a2 in LSeg (|[a,c]|,|[a,d]|) ) assume a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } ; ::_thesis: a2 in LSeg (|[a,c]|,|[a,d]|) then consider p being Point of (TOP-REAL 2) such that A8: a2 = p and A9: p `1 = a and A10: p `2 <= d and A11: p `2 >= c ; now__::_thesis:_(_(_d_<>_c_&_a2_in_LSeg_(|[a,c]|,|[a,d]|)_)_or_(_d_=_c_&_a2_in_LSeg_(|[a,c]|,|[a,d]|)_)_) percases ( d <> c or d = c ) ; caseA12: d <> c ; ::_thesis: a2 in LSeg (|[a,c]|,|[a,d]|) reconsider lambda = ((p `2) - c) / (d - c) as Real ; d >= c by A10, A11, XXREAL_0:2; then d > c by A12, XXREAL_0:1; then A13: d - c > 0 by XREAL_1:50; A14: (p `2) - c >= 0 by A11, XREAL_1:48; d - c >= (p `2) - c by A10, XREAL_1:9; then (d - c) / (d - c) >= ((p `2) - c) / (d - c) by A13, XREAL_1:72; then A15: 1 >= lambda by A13, XCMPLX_1:60; A16: ((1 - lambda) * c) + (lambda * d) = ((((d - c) / (d - c)) - (((p `2) - c) / (d - c))) * c) + ((((p `2) - c) / (d - c)) * d) by A13, XCMPLX_1:60 .= ((((d - c) - ((p `2) - c)) / (d - c)) * c) + ((((p `2) - c) / (d - c)) * d) by XCMPLX_1:120 .= (c * ((d - (p `2)) / (d - c))) + ((d * ((p `2) - c)) / (d - c)) by XCMPLX_1:74 .= ((c * (d - (p `2))) / (d - c)) + ((d * ((p `2) - c)) / (d - c)) by XCMPLX_1:74 .= (((c * d) - (c * (p `2))) + ((d * (p `2)) - (d * c))) / (d - c) by XCMPLX_1:62 .= ((d - c) * (p `2)) / (d - c) .= (p `2) * ((d - c) / (d - c)) by XCMPLX_1:74 .= (p `2) * 1 by A13, XCMPLX_1:60 .= p `2 ; ((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|) = |[((1 - lambda) * a),((1 - lambda) * c)]| + (lambda * |[a,d]|) by EUCLID:58 .= |[((1 - lambda) * a),((1 - lambda) * c)]| + |[(lambda * a),(lambda * d)]| by EUCLID:58 .= |[(((1 - lambda) * a) + (lambda * a)),(((1 - lambda) * c) + (lambda * d))]| by EUCLID:56 .= p by A9, A16, EUCLID:53 ; hence a2 in LSeg (|[a,c]|,|[a,d]|) by A8, A13, A14, A15; ::_thesis: verum end; case d = c ; ::_thesis: a2 in LSeg (|[a,c]|,|[a,d]|) then A17: p `2 = c by A10, A11, XXREAL_0:1; reconsider lambda = 0 as Real ; ((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|) = |[((1 - lambda) * a),((1 - lambda) * c)]| + (lambda * |[a,d]|) by EUCLID:58 .= |[((1 - lambda) * a),((1 - lambda) * c)]| + |[(lambda * a),(lambda * d)]| by EUCLID:58 .= |[(((1 - lambda) * a) + (lambda * a)),(((1 - lambda) * c) + (lambda * d))]| by EUCLID:56 .= p by A9, A17, EUCLID:53 ; hence a2 in LSeg (|[a,c]|,|[a,d]|) by A8; ::_thesis: verum end; end; end; hence a2 in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: verum end; LSeg (|[a,c]|,|[a,d]|) c= { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } proof let a2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not a2 in LSeg (|[a,c]|,|[a,d]|) or a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } ) assume a2 in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } then consider lambda being Real such that A18: a2 = ((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|) and A19: 0 <= lambda and A20: lambda <= 1 ; set q = ((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|); A21: (((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|)) `1 = (((1 - lambda) * |[a,c]|) `1) + ((lambda * |[a,d]|) `1) by TOPREAL3:2 .= ((1 - lambda) * (|[a,c]| `1)) + ((lambda * |[a,d]|) `1) by TOPREAL3:4 .= ((1 - lambda) * (|[a,c]| `1)) + (lambda * (|[a,d]| `1)) by TOPREAL3:4 .= ((1 - lambda) * a) + (lambda * a) by A3, EUCLID:52 .= a ; A22: (((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|)) `2 = (((1 - lambda) * |[a,c]|) `2) + ((lambda * |[a,d]|) `2) by TOPREAL3:2 .= ((1 - lambda) * (|[a,c]| `2)) + ((lambda * |[a,d]|) `2) by TOPREAL3:4 .= ((1 - lambda) * (|[a,c]| `2)) + (lambda * (|[a,d]| `2)) by TOPREAL3:4 .= ((1 - lambda) * c) + (lambda * d) by A4, EUCLID:52 ; then A23: (((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|)) `2 <= d by A2, A20, XREAL_1:172; (((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|)) `2 >= c by A2, A19, A20, A22, XREAL_1:173; hence a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } by A18, A21, A23; ::_thesis: verum end; hence { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } = LSeg (|[a,c]|,|[a,d]|) by A7, XBOOLE_0:def_10; ::_thesis: ( LSeg (|[a,d]|,|[b,d]|) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) } & LSeg (|[a,c]|,|[b,c]|) = { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 <= b & q1 `1 >= a & q1 `2 = c ) } & LSeg (|[b,c]|,|[b,d]|) = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = b & q2 `2 <= d & q2 `2 >= c ) } ) A24: { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } c= LSeg (|[a,d]|,|[b,d]|) proof let a2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } or a2 in LSeg (|[a,d]|,|[b,d]|) ) assume a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } ; ::_thesis: a2 in LSeg (|[a,d]|,|[b,d]|) then consider p being Point of (TOP-REAL 2) such that A25: a2 = p and A26: p `1 <= b and A27: p `1 >= a and A28: p `2 = d ; now__::_thesis:_(_(_b_<>_a_&_a2_in_LSeg_(|[a,d]|,|[b,d]|)_)_or_(_b_=_a_&_a2_in_LSeg_(|[a,d]|,|[b,d]|)_)_) percases ( b <> a or b = a ) ; caseA29: b <> a ; ::_thesis: a2 in LSeg (|[a,d]|,|[b,d]|) reconsider lambda = ((p `1) - a) / (b - a) as Real ; b >= a by A26, A27, XXREAL_0:2; then b > a by A29, XXREAL_0:1; then A30: b - a > 0 by XREAL_1:50; A31: (p `1) - a >= 0 by A27, XREAL_1:48; b - a >= (p `1) - a by A26, XREAL_1:9; then (b - a) / (b - a) >= ((p `1) - a) / (b - a) by A30, XREAL_1:72; then A32: 1 >= lambda by A30, XCMPLX_1:60; A33: ((1 - lambda) * a) + (lambda * b) = ((((b - a) / (b - a)) - (((p `1) - a) / (b - a))) * a) + ((((p `1) - a) / (b - a)) * b) by A30, XCMPLX_1:60 .= ((((b - a) - ((p `1) - a)) / (b - a)) * a) + ((((p `1) - a) / (b - a)) * b) by XCMPLX_1:120 .= (a * ((b - (p `1)) / (b - a))) + ((b * ((p `1) - a)) / (b - a)) by XCMPLX_1:74 .= ((a * (b - (p `1))) / (b - a)) + ((b * ((p `1) - a)) / (b - a)) by XCMPLX_1:74 .= (((a * b) - (a * (p `1))) + ((b * (p `1)) - (b * a))) / (b - a) by XCMPLX_1:62 .= ((b - a) * (p `1)) / (b - a) .= (p `1) * ((b - a) / (b - a)) by XCMPLX_1:74 .= (p `1) * 1 by A30, XCMPLX_1:60 .= p `1 ; ((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|) = |[((1 - lambda) * a),((1 - lambda) * d)]| + (lambda * |[b,d]|) by EUCLID:58 .= |[((1 - lambda) * a),((1 - lambda) * d)]| + |[(lambda * b),(lambda * d)]| by EUCLID:58 .= |[(((1 - lambda) * a) + (lambda * b)),(((1 - lambda) * d) + (lambda * d))]| by EUCLID:56 .= p by A28, A33, EUCLID:53 ; hence a2 in LSeg (|[a,d]|,|[b,d]|) by A25, A30, A31, A32; ::_thesis: verum end; case b = a ; ::_thesis: a2 in LSeg (|[a,d]|,|[b,d]|) then A34: p `1 = a by A26, A27, XXREAL_0:1; reconsider lambda = 0 as Real ; ((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|) = |[((1 - lambda) * a),((1 - lambda) * d)]| + (lambda * |[b,d]|) by EUCLID:58 .= |[((1 - lambda) * a),((1 - lambda) * d)]| + |[(lambda * b),(lambda * d)]| by EUCLID:58 .= |[(((1 - lambda) * a) + (lambda * b)),(((1 - lambda) * d) + (lambda * d))]| by EUCLID:56 .= p by A28, A34, EUCLID:53 ; hence a2 in LSeg (|[a,d]|,|[b,d]|) by A25; ::_thesis: verum end; end; end; hence a2 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: verum end; LSeg (|[a,d]|,|[b,d]|) c= { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } proof let a2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not a2 in LSeg (|[a,d]|,|[b,d]|) or a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } ) assume a2 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } then consider lambda being Real such that A35: a2 = ((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|) and A36: 0 <= lambda and A37: lambda <= 1 ; set q = ((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|); A38: (((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|)) `2 = (((1 - lambda) * |[a,d]|) `2) + ((lambda * |[b,d]|) `2) by TOPREAL3:2 .= ((1 - lambda) * (|[a,d]| `2)) + ((lambda * |[b,d]|) `2) by TOPREAL3:4 .= ((1 - lambda) * (|[a,d]| `2)) + (lambda * (|[b,d]| `2)) by TOPREAL3:4 .= ((1 - lambda) * d) + (lambda * d) by A4, EUCLID:52 .= d ; A39: (((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|)) `1 = (((1 - lambda) * |[a,d]|) `1) + ((lambda * |[b,d]|) `1) by TOPREAL3:2 .= ((1 - lambda) * (|[a,d]| `1)) + ((lambda * |[b,d]|) `1) by TOPREAL3:4 .= ((1 - lambda) * (|[a,d]| `1)) + (lambda * (|[b,d]| `1)) by TOPREAL3:4 .= ((1 - lambda) * a) + (lambda * b) by A3, EUCLID:52 ; then A40: (((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|)) `1 <= b by A1, A37, XREAL_1:172; (((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|)) `1 >= a by A1, A36, A37, A39, XREAL_1:173; hence a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } by A35, A38, A40; ::_thesis: verum end; hence { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } = LSeg (|[a,d]|,|[b,d]|) by A24, XBOOLE_0:def_10; ::_thesis: ( LSeg (|[a,c]|,|[b,c]|) = { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 <= b & q1 `1 >= a & q1 `2 = c ) } & LSeg (|[b,c]|,|[b,d]|) = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = b & q2 `2 <= d & q2 `2 >= c ) } ) A41: { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } c= LSeg (|[a,c]|,|[b,c]|) proof let a2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } or a2 in LSeg (|[a,c]|,|[b,c]|) ) assume a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } ; ::_thesis: a2 in LSeg (|[a,c]|,|[b,c]|) then consider p being Point of (TOP-REAL 2) such that A42: a2 = p and A43: p `1 <= b and A44: p `1 >= a and A45: p `2 = c ; now__::_thesis:_(_(_b_<>_a_&_a2_in_LSeg_(|[a,c]|,|[b,c]|)_)_or_(_b_=_a_&_a2_in_LSeg_(|[a,c]|,|[b,c]|)_)_) percases ( b <> a or b = a ) ; caseA46: b <> a ; ::_thesis: a2 in LSeg (|[a,c]|,|[b,c]|) reconsider lambda = ((p `1) - a) / (b - a) as Real ; b >= a by A43, A44, XXREAL_0:2; then b > a by A46, XXREAL_0:1; then A47: b - a > 0 by XREAL_1:50; A48: (p `1) - a >= 0 by A44, XREAL_1:48; b - a >= (p `1) - a by A43, XREAL_1:9; then (b - a) / (b - a) >= ((p `1) - a) / (b - a) by A47, XREAL_1:72; then A49: 1 >= lambda by A47, XCMPLX_1:60; A50: ((1 - lambda) * a) + (lambda * b) = ((((b - a) / (b - a)) - (((p `1) - a) / (b - a))) * a) + ((((p `1) - a) / (b - a)) * b) by A47, XCMPLX_1:60 .= ((((b - a) - ((p `1) - a)) / (b - a)) * a) + ((((p `1) - a) / (b - a)) * b) by XCMPLX_1:120 .= (a * ((b - (p `1)) / (b - a))) + ((b * ((p `1) - a)) / (b - a)) by XCMPLX_1:74 .= ((a * (b - (p `1))) / (b - a)) + ((b * ((p `1) - a)) / (b - a)) by XCMPLX_1:74 .= (((a * b) - (a * (p `1))) + ((b * (p `1)) - (b * a))) / (b - a) by XCMPLX_1:62 .= ((b - a) * (p `1)) / (b - a) .= (p `1) * ((b - a) / (b - a)) by XCMPLX_1:74 .= (p `1) * 1 by A47, XCMPLX_1:60 .= p `1 ; ((1 - lambda) * |[a,c]|) + (lambda * |[b,c]|) = |[((1 - lambda) * a),((1 - lambda) * c)]| + (lambda * |[b,c]|) by EUCLID:58 .= |[((1 - lambda) * a),((1 - lambda) * c)]| + |[(lambda * b),(lambda * c)]| by EUCLID:58 .= |[(((1 - lambda) * a) + (lambda * b)),(((1 - lambda) * c) + (lambda * c))]| by EUCLID:56 .= p by A45, A50, EUCLID:53 ; hence a2 in LSeg (|[a,c]|,|[b,c]|) by A42, A47, A48, A49; ::_thesis: verum end; case b = a ; ::_thesis: a2 in LSeg (|[a,c]|,|[b,c]|) then A51: p `1 = a by A43, A44, XXREAL_0:1; reconsider lambda = 0 as Real ; ((1 - lambda) * |[a,c]|) + (lambda * |[b,c]|) = |[((1 - lambda) * a),((1 - lambda) * c)]| + (lambda * |[b,c]|) by EUCLID:58 .= |[((1 - lambda) * a),((1 - lambda) * c)]| + |[(lambda * b),(lambda * c)]| by EUCLID:58 .= |[(((1 - lambda) * a) + (lambda * b)),(((1 - lambda) * c) + (lambda * c))]| by EUCLID:56 .= p by A45, A51, EUCLID:53 ; hence a2 in LSeg (|[a,c]|,|[b,c]|) by A42; ::_thesis: verum end; end; end; hence a2 in LSeg (|[a,c]|,|[b,c]|) ; ::_thesis: verum end; LSeg (|[a,c]|,|[b,c]|) c= { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } proof let a2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not a2 in LSeg (|[a,c]|,|[b,c]|) or a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } ) assume a2 in LSeg (|[a,c]|,|[b,c]|) ; ::_thesis: a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } then consider lambda being Real such that A52: a2 = ((1 - lambda) * |[a,c]|) + (lambda * |[b,c]|) and A53: 0 <= lambda and A54: lambda <= 1 ; set q = ((1 - lambda) * |[a,c]|) + (lambda * |[b,c]|); A55: (((1 - lambda) * |[a,c]|) + (lambda * |[b,c]|)) `2 = (((1 - lambda) * |[a,c]|) `2) + ((lambda * |[b,c]|) `2) by TOPREAL3:2 .= ((1 - lambda) * (|[a,c]| `2)) + ((lambda * |[b,c]|) `2) by TOPREAL3:4 .= ((1 - lambda) * (|[a,c]| `2)) + (lambda * (|[b,c]| `2)) by TOPREAL3:4 .= ((1 - lambda) * c) + (lambda * c) by A6, EUCLID:52 .= c ; A56: (((1 - lambda) * |[a,c]|) + (lambda * |[b,c]|)) `1 = (((1 - lambda) * |[a,c]|) `1) + ((lambda * |[b,c]|) `1) by TOPREAL3:2 .= ((1 - lambda) * (|[a,c]| `1)) + ((lambda * |[b,c]|) `1) by TOPREAL3:4 .= ((1 - lambda) * (|[a,c]| `1)) + (lambda * (|[b,c]| `1)) by TOPREAL3:4 .= ((1 - lambda) * a) + (lambda * b) by A5, EUCLID:52 ; then A57: (((1 - lambda) * |[a,c]|) + (lambda * |[b,c]|)) `1 <= b by A1, A54, XREAL_1:172; (((1 - lambda) * |[a,c]|) + (lambda * |[b,c]|)) `1 >= a by A1, A53, A54, A56, XREAL_1:173; hence a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } by A52, A55, A57; ::_thesis: verum end; hence { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } = LSeg (|[a,c]|,|[b,c]|) by A41, XBOOLE_0:def_10; ::_thesis: LSeg (|[b,c]|,|[b,d]|) = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = b & q2 `2 <= d & q2 `2 >= c ) } A58: { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } c= LSeg (|[b,c]|,|[b,d]|) proof let a2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } or a2 in LSeg (|[b,c]|,|[b,d]|) ) assume a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } ; ::_thesis: a2 in LSeg (|[b,c]|,|[b,d]|) then consider p being Point of (TOP-REAL 2) such that A59: a2 = p and A60: p `1 = b and A61: p `2 <= d and A62: p `2 >= c ; now__::_thesis:_(_(_d_<>_c_&_a2_in_LSeg_(|[b,c]|,|[b,d]|)_)_or_(_d_=_c_&_a2_in_LSeg_(|[b,c]|,|[b,d]|)_)_) percases ( d <> c or d = c ) ; caseA63: d <> c ; ::_thesis: a2 in LSeg (|[b,c]|,|[b,d]|) reconsider lambda = ((p `2) - c) / (d - c) as Real ; d >= c by A61, A62, XXREAL_0:2; then d > c by A63, XXREAL_0:1; then A64: d - c > 0 by XREAL_1:50; A65: (p `2) - c >= 0 by A62, XREAL_1:48; d - c >= (p `2) - c by A61, XREAL_1:9; then (d - c) / (d - c) >= ((p `2) - c) / (d - c) by A64, XREAL_1:72; then A66: 1 >= lambda by A64, XCMPLX_1:60; A67: ((1 - lambda) * c) + (lambda * d) = ((((d - c) / (d - c)) - (((p `2) - c) / (d - c))) * c) + ((((p `2) - c) / (d - c)) * d) by A64, XCMPLX_1:60 .= ((((d - c) - ((p `2) - c)) / (d - c)) * c) + ((((p `2) - c) / (d - c)) * d) by XCMPLX_1:120 .= (c * ((d - (p `2)) / (d - c))) + ((d * ((p `2) - c)) / (d - c)) by XCMPLX_1:74 .= ((c * (d - (p `2))) / (d - c)) + ((d * ((p `2) - c)) / (d - c)) by XCMPLX_1:74 .= (((c * d) - (c * (p `2))) + ((d * (p `2)) - (d * c))) / (d - c) by XCMPLX_1:62 .= ((d - c) * (p `2)) / (d - c) .= (p `2) * ((d - c) / (d - c)) by XCMPLX_1:74 .= (p `2) * 1 by A64, XCMPLX_1:60 .= p `2 ; ((1 - lambda) * |[b,c]|) + (lambda * |[b,d]|) = |[((1 - lambda) * b),((1 - lambda) * c)]| + (lambda * |[b,d]|) by EUCLID:58 .= |[((1 - lambda) * b),((1 - lambda) * c)]| + |[(lambda * b),(lambda * d)]| by EUCLID:58 .= |[(((1 - lambda) * b) + (lambda * b)),(((1 - lambda) * c) + (lambda * d))]| by EUCLID:56 .= p by A60, A67, EUCLID:53 ; hence a2 in LSeg (|[b,c]|,|[b,d]|) by A59, A64, A65, A66; ::_thesis: verum end; case d = c ; ::_thesis: a2 in LSeg (|[b,c]|,|[b,d]|) then A68: p `2 = c by A61, A62, XXREAL_0:1; reconsider lambda = 0 as Real ; ((1 - lambda) * |[b,c]|) + (lambda * |[b,d]|) = |[((1 - lambda) * b),((1 - lambda) * c)]| + (lambda * |[b,d]|) by EUCLID:58 .= |[((1 - lambda) * b),((1 - lambda) * c)]| + |[(lambda * b),(lambda * d)]| by EUCLID:58 .= |[(((1 - lambda) * b) + (lambda * b)),(((1 - lambda) * c) + (lambda * d))]| by EUCLID:56 .= p by A60, A68, EUCLID:53 ; hence a2 in LSeg (|[b,c]|,|[b,d]|) by A59; ::_thesis: verum end; end; end; hence a2 in LSeg (|[b,c]|,|[b,d]|) ; ::_thesis: verum end; LSeg (|[b,c]|,|[b,d]|) c= { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } proof let a2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not a2 in LSeg (|[b,c]|,|[b,d]|) or a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } ) assume a2 in LSeg (|[b,c]|,|[b,d]|) ; ::_thesis: a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } then consider lambda being Real such that A69: a2 = ((1 - lambda) * |[b,c]|) + (lambda * |[b,d]|) and A70: 0 <= lambda and A71: lambda <= 1 ; set q = ((1 - lambda) * |[b,c]|) + (lambda * |[b,d]|); A72: (((1 - lambda) * |[b,c]|) + (lambda * |[b,d]|)) `1 = (((1 - lambda) * |[b,c]|) `1) + ((lambda * |[b,d]|) `1) by TOPREAL3:2 .= ((1 - lambda) * (|[b,c]| `1)) + ((lambda * |[b,d]|) `1) by TOPREAL3:4 .= ((1 - lambda) * (|[b,c]| `1)) + (lambda * (|[b,d]| `1)) by TOPREAL3:4 .= ((1 - lambda) * b) + (lambda * b) by A5, EUCLID:52 .= b ; A73: (((1 - lambda) * |[b,c]|) + (lambda * |[b,d]|)) `2 = (((1 - lambda) * |[b,c]|) `2) + ((lambda * |[b,d]|) `2) by TOPREAL3:2 .= ((1 - lambda) * (|[b,c]| `2)) + ((lambda * |[b,d]|) `2) by TOPREAL3:4 .= ((1 - lambda) * (|[b,c]| `2)) + (lambda * (|[b,d]| `2)) by TOPREAL3:4 .= ((1 - lambda) * c) + (lambda * d) by A6, EUCLID:52 ; then A74: (((1 - lambda) * |[b,c]|) + (lambda * |[b,d]|)) `2 <= d by A2, A71, XREAL_1:172; (((1 - lambda) * |[b,c]|) + (lambda * |[b,d]|)) `2 >= c by A2, A70, A71, A73, XREAL_1:173; hence a2 in { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } by A69, A72, A74; ::_thesis: verum end; hence { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } = LSeg (|[b,c]|,|[b,d]|) by A58, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th31: :: JGRAPH_6:31 for a, b, c, d being real number st a <= b & c <= d holds (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) = {|[a,c]|} proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) = {|[a,c]|} ) assume that A1: a <= b and A2: c <= d ; ::_thesis: (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) = {|[a,c]|} for ax being set holds ( ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) iff ax = |[a,c]| ) proof let ax be set ; ::_thesis: ( ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) iff ax = |[a,c]| ) thus ( ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) implies ax = |[a,c]| ) ::_thesis: ( ax = |[a,c]| implies ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) ) proof assume A3: ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) ; ::_thesis: ax = |[a,c]| then A4: ax in LSeg (|[a,c]|,|[a,d]|) by XBOOLE_0:def_4; ax in LSeg (|[a,c]|,|[b,c]|) by A3, XBOOLE_0:def_4; then ax in { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) } by A1, Th30; then A5: ex p2 being Point of (TOP-REAL 2) st ( p2 = ax & p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) ; ax in { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = a & p2 `2 <= d & p2 `2 >= c ) } by A2, A4, Th30; then ex p being Point of (TOP-REAL 2) st ( p = ax & p `1 = a & p `2 <= d & p `2 >= c ) ; hence ax = |[a,c]| by A5, EUCLID:53; ::_thesis: verum end; assume A6: ax = |[a,c]| ; ::_thesis: ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) then A7: ax in LSeg (|[a,c]|,|[a,d]|) by RLTOPSP1:68; ax in LSeg (|[a,c]|,|[b,c]|) by A6, RLTOPSP1:68; hence ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) by A7, XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) = {|[a,c]|} by TARSKI:def_1; ::_thesis: verum end; theorem Th32: :: JGRAPH_6:32 for a, b, c, d being real number st a <= b & c <= d holds (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,c]|} proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,c]|} ) assume that A1: a <= b and A2: c <= d ; ::_thesis: (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,c]|} for ax being set holds ( ax in (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) iff ax = |[b,c]| ) proof let ax be set ; ::_thesis: ( ax in (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) iff ax = |[b,c]| ) thus ( ax in (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) implies ax = |[b,c]| ) ::_thesis: ( ax = |[b,c]| implies ax in (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) ) proof assume A3: ax in (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) ; ::_thesis: ax = |[b,c]| then A4: ax in LSeg (|[a,c]|,|[b,c]|) by XBOOLE_0:def_4; A5: ax in LSeg (|[b,c]|,|[b,d]|) by A3, XBOOLE_0:def_4; ax in { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 <= b & q1 `1 >= a & q1 `2 = c ) } by A1, A4, Th30; then A6: ex p2 being Point of (TOP-REAL 2) st ( p2 = ax & p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) ; ax in { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = b & q2 `2 <= d & q2 `2 >= c ) } by A2, A5, Th30; then ex p being Point of (TOP-REAL 2) st ( p = ax & p `1 = b & p `2 <= d & p `2 >= c ) ; hence ax = |[b,c]| by A6, EUCLID:53; ::_thesis: verum end; assume A7: ax = |[b,c]| ; ::_thesis: ax in (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) then A8: ax in LSeg (|[a,c]|,|[b,c]|) by RLTOPSP1:68; ax in LSeg (|[b,c]|,|[b,d]|) by A7, RLTOPSP1:68; hence ax in (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) by A8, XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,c]|} by TARSKI:def_1; ::_thesis: verum end; theorem Th33: :: JGRAPH_6:33 for a, b, c, d being real number st a <= b & c <= d holds (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,d]|} proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,d]|} ) assume that A1: a <= b and A2: c <= d ; ::_thesis: (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,d]|} for ax being set holds ( ax in (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) iff ax = |[b,d]| ) proof let ax be set ; ::_thesis: ( ax in (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) iff ax = |[b,d]| ) thus ( ax in (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) implies ax = |[b,d]| ) ::_thesis: ( ax = |[b,d]| implies ax in (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) ) proof assume A3: ax in (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) ; ::_thesis: ax = |[b,d]| then A4: ax in LSeg (|[b,c]|,|[b,d]|) by XBOOLE_0:def_4; ax in LSeg (|[a,d]|,|[b,d]|) by A3, XBOOLE_0:def_4; then ax in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } by A1, Th30; then A5: ex p being Point of (TOP-REAL 2) st ( p = ax & p `1 <= b & p `1 >= a & p `2 = d ) ; ax in { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } by A2, A4, Th30; then ex p2 being Point of (TOP-REAL 2) st ( p2 = ax & p2 `1 = b & p2 `2 <= d & p2 `2 >= c ) ; hence ax = |[b,d]| by A5, EUCLID:53; ::_thesis: verum end; assume A6: ax = |[b,d]| ; ::_thesis: ax in (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) then A7: ax in LSeg (|[a,d]|,|[b,d]|) by RLTOPSP1:68; ax in LSeg (|[b,c]|,|[b,d]|) by A6, RLTOPSP1:68; hence ax in (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) by A7, XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,d]|} by TARSKI:def_1; ::_thesis: verum end; theorem Th34: :: JGRAPH_6:34 for a, b, c, d being real number st a <= b & c <= d holds (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) = {|[a,d]|} proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) = {|[a,d]|} ) assume that A1: a <= b and A2: c <= d ; ::_thesis: (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) = {|[a,d]|} for ax being set holds ( ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) iff ax = |[a,d]| ) proof let ax be set ; ::_thesis: ( ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) iff ax = |[a,d]| ) thus ( ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) implies ax = |[a,d]| ) ::_thesis: ( ax = |[a,d]| implies ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) ) proof assume A3: ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) ; ::_thesis: ax = |[a,d]| then A4: ax in LSeg (|[a,c]|,|[a,d]|) by XBOOLE_0:def_4; ax in LSeg (|[a,d]|,|[b,d]|) by A3, XBOOLE_0:def_4; then ax in { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) } by A1, Th30; then A5: ex p2 being Point of (TOP-REAL 2) st ( p2 = ax & p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) ; ax in { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = a & p2 `2 <= d & p2 `2 >= c ) } by A2, A4, Th30; then ex p being Point of (TOP-REAL 2) st ( p = ax & p `1 = a & p `2 <= d & p `2 >= c ) ; hence ax = |[a,d]| by A5, EUCLID:53; ::_thesis: verum end; assume A6: ax = |[a,d]| ; ::_thesis: ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) then A7: ax in LSeg (|[a,c]|,|[a,d]|) by RLTOPSP1:68; ax in LSeg (|[a,d]|,|[b,d]|) by A6, RLTOPSP1:68; hence ax in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) by A7, XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) = {|[a,d]|} by TARSKI:def_1; ::_thesis: verum end; theorem Th35: :: JGRAPH_6:35 { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } proof thus { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } c= { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } c= { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } or x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } ) assume x in { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } then ex q being Point of (TOP-REAL 2) st ( x = q & ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) ) ; hence x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } ; ::_thesis: verum end; thus { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } c= { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } or x in { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } ) assume x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } ; ::_thesis: x in { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) ) ; hence x in { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } ; ::_thesis: verum end; end; theorem Th36: :: JGRAPH_6:36 for a, b, c, d being real number st a <= b & c <= d holds W-bound (rectangle (a,b,c,d)) = a proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies W-bound (rectangle (a,b,c,d)) = a ) assume that A1: a <= b and A2: c <= d ; ::_thesis: W-bound (rectangle (a,b,c,d)) = a set X = rectangle (a,b,c,d); reconsider Z = (proj1 | (rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) as Subset of REAL ; A3: rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) by PRE_TOPC:8; A4: for p being real number st p in Z holds p >= a proof let p be real number ; ::_thesis: ( p in Z implies p >= a ) assume p in Z ; ::_thesis: p >= a then consider p0 being set such that A5: p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and A6: p = (proj1 | (rectangle (a,b,c,d))) . p0 by FUNCT_2:64; reconsider p0 = p0 as Point of (TOP-REAL 2) by A3, A5; rectangle (a,b,c,d) = { q where q is Point of (TOP-REAL 2) : ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) } by A1, A2, SPPOL_2:54; then ex q being Point of (TOP-REAL 2) st ( p0 = q & ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) ) by A3, A5; hence p >= a by A1, A3, A5, A6, PSCOMP_1:22; ::_thesis: verum end; A7: for q being real number st ( for p being real number st p in Z holds p >= q ) holds a >= q proof let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds p >= q ) implies a >= q ) assume A8: for p being real number st p in Z holds p >= q ; ::_thesis: a >= q |[a,c]| in LSeg (|[a,c]|,|[b,c]|) by RLTOPSP1:68; then A9: |[a,c]| in (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by XBOOLE_0:def_3; rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3; then A10: |[a,c]| in rectangle (a,b,c,d) by A9, XBOOLE_0:def_3; then (proj1 | (rectangle (a,b,c,d))) . |[a,c]| = |[a,c]| `1 by PSCOMP_1:22 .= a by EUCLID:52 ; hence a >= q by A3, A8, A10, FUNCT_2:35; ::_thesis: verum end; thus W-bound (rectangle (a,b,c,d)) = lower_bound (proj1 | (rectangle (a,b,c,d))) by PSCOMP_1:def_7 .= lower_bound Z by PSCOMP_1:def_1 .= a by A4, A7, SEQ_4:44 ; ::_thesis: verum end; theorem Th37: :: JGRAPH_6:37 for a, b, c, d being real number st a <= b & c <= d holds N-bound (rectangle (a,b,c,d)) = d proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies N-bound (rectangle (a,b,c,d)) = d ) assume that A1: a <= b and A2: c <= d ; ::_thesis: N-bound (rectangle (a,b,c,d)) = d set X = rectangle (a,b,c,d); reconsider Z = (proj2 | (rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) as Subset of REAL ; A3: rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) by PRE_TOPC:8; A4: for p being real number st p in Z holds p <= d proof let p be real number ; ::_thesis: ( p in Z implies p <= d ) assume p in Z ; ::_thesis: p <= d then consider p0 being set such that A5: p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and A6: p = (proj2 | (rectangle (a,b,c,d))) . p0 by FUNCT_2:64; reconsider p0 = p0 as Point of (TOP-REAL 2) by A3, A5; rectangle (a,b,c,d) = { q where q is Point of (TOP-REAL 2) : ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) } by A1, A2, SPPOL_2:54; then ex q being Point of (TOP-REAL 2) st ( p0 = q & ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) ) by A3, A5; hence p <= d by A2, A3, A5, A6, PSCOMP_1:23; ::_thesis: verum end; A7: for q being real number st ( for p being real number st p in Z holds p <= q ) holds d <= q proof let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds p <= q ) implies d <= q ) assume A8: for p being real number st p in Z holds p <= q ; ::_thesis: d <= q |[b,d]| in LSeg (|[b,c]|,|[b,d]|) by RLTOPSP1:68; then A9: |[b,d]| in (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by XBOOLE_0:def_3; rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3; then A10: |[b,d]| in rectangle (a,b,c,d) by A9, XBOOLE_0:def_3; then (proj2 | (rectangle (a,b,c,d))) . |[b,d]| = |[b,d]| `2 by PSCOMP_1:23 .= d by EUCLID:52 ; hence d <= q by A3, A8, A10, FUNCT_2:35; ::_thesis: verum end; thus N-bound (rectangle (a,b,c,d)) = upper_bound (proj2 | (rectangle (a,b,c,d))) by PSCOMP_1:def_8 .= upper_bound Z by PSCOMP_1:def_2 .= d by A4, A7, SEQ_4:46 ; ::_thesis: verum end; theorem Th38: :: JGRAPH_6:38 for a, b, c, d being real number st a <= b & c <= d holds E-bound (rectangle (a,b,c,d)) = b proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies E-bound (rectangle (a,b,c,d)) = b ) assume that A1: a <= b and A2: c <= d ; ::_thesis: E-bound (rectangle (a,b,c,d)) = b set X = rectangle (a,b,c,d); reconsider Z = (proj1 | (rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) as Subset of REAL ; A3: rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) by PRE_TOPC:8; A4: for p being real number st p in Z holds p <= b proof let p be real number ; ::_thesis: ( p in Z implies p <= b ) assume p in Z ; ::_thesis: p <= b then consider p0 being set such that A5: p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and A6: p = (proj1 | (rectangle (a,b,c,d))) . p0 by FUNCT_2:64; reconsider p0 = p0 as Point of (TOP-REAL 2) by A3, A5; rectangle (a,b,c,d) = { q where q is Point of (TOP-REAL 2) : ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) } by A1, A2, SPPOL_2:54; then ex q being Point of (TOP-REAL 2) st ( p0 = q & ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) ) by A3, A5; hence p <= b by A1, A3, A5, A6, PSCOMP_1:22; ::_thesis: verum end; A7: for q being real number st ( for p being real number st p in Z holds p <= q ) holds b <= q proof let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds p <= q ) implies b <= q ) assume A8: for p being real number st p in Z holds p <= q ; ::_thesis: b <= q |[b,d]| in LSeg (|[b,c]|,|[b,d]|) by RLTOPSP1:68; then A9: |[b,d]| in (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by XBOOLE_0:def_3; rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3; then A10: |[b,d]| in rectangle (a,b,c,d) by A9, XBOOLE_0:def_3; then (proj1 | (rectangle (a,b,c,d))) . |[b,d]| = |[b,d]| `1 by PSCOMP_1:22 .= b by EUCLID:52 ; hence b <= q by A3, A8, A10, FUNCT_2:35; ::_thesis: verum end; thus E-bound (rectangle (a,b,c,d)) = upper_bound (proj1 | (rectangle (a,b,c,d))) by PSCOMP_1:def_9 .= upper_bound Z by PSCOMP_1:def_2 .= b by A4, A7, SEQ_4:46 ; ::_thesis: verum end; theorem Th39: :: JGRAPH_6:39 for a, b, c, d being real number st a <= b & c <= d holds S-bound (rectangle (a,b,c,d)) = c proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies S-bound (rectangle (a,b,c,d)) = c ) assume that A1: a <= b and A2: c <= d ; ::_thesis: S-bound (rectangle (a,b,c,d)) = c set X = rectangle (a,b,c,d); reconsider Z = (proj2 | (rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) as Subset of REAL ; A3: rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) by PRE_TOPC:8; A4: for p being real number st p in Z holds p >= c proof let p be real number ; ::_thesis: ( p in Z implies p >= c ) assume p in Z ; ::_thesis: p >= c then consider p0 being set such that A5: p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and p0 in the carrier of ((TOP-REAL 2) | (rectangle (a,b,c,d))) and A6: p = (proj2 | (rectangle (a,b,c,d))) . p0 by FUNCT_2:64; reconsider p0 = p0 as Point of (TOP-REAL 2) by A3, A5; rectangle (a,b,c,d) = { q where q is Point of (TOP-REAL 2) : ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) } by A1, A2, SPPOL_2:54; then ex q being Point of (TOP-REAL 2) st ( p0 = q & ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) ) by A3, A5; hence p >= c by A2, A3, A5, A6, PSCOMP_1:23; ::_thesis: verum end; A7: for q being real number st ( for p being real number st p in Z holds p >= q ) holds c >= q proof let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds p >= q ) implies c >= q ) assume A8: for p being real number st p in Z holds p >= q ; ::_thesis: c >= q |[b,c]| in LSeg (|[b,c]|,|[b,d]|) by RLTOPSP1:68; then A9: |[b,c]| in (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by XBOOLE_0:def_3; rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3; then A10: |[b,c]| in rectangle (a,b,c,d) by A9, XBOOLE_0:def_3; then (proj2 | (rectangle (a,b,c,d))) . |[b,c]| = |[b,c]| `2 by PSCOMP_1:23 .= c by EUCLID:52 ; hence c >= q by A3, A8, A10, FUNCT_2:35; ::_thesis: verum end; thus S-bound (rectangle (a,b,c,d)) = lower_bound (proj2 | (rectangle (a,b,c,d))) by PSCOMP_1:def_10 .= lower_bound Z by PSCOMP_1:def_1 .= c by A4, A7, SEQ_4:44 ; ::_thesis: verum end; theorem Th40: :: JGRAPH_6:40 for a, b, c, d being real number st a <= b & c <= d holds NW-corner (rectangle (a,b,c,d)) = |[a,d]| proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies NW-corner (rectangle (a,b,c,d)) = |[a,d]| ) assume that A1: a <= b and A2: c <= d ; ::_thesis: NW-corner (rectangle (a,b,c,d)) = |[a,d]| set K = rectangle (a,b,c,d); A3: NW-corner (rectangle (a,b,c,d)) = |[(W-bound (rectangle (a,b,c,d))),(N-bound (rectangle (a,b,c,d)))]| by PSCOMP_1:def_12; W-bound (rectangle (a,b,c,d)) = a by A1, A2, Th36; hence NW-corner (rectangle (a,b,c,d)) = |[a,d]| by A1, A2, A3, Th37; ::_thesis: verum end; theorem Th41: :: JGRAPH_6:41 for a, b, c, d being real number st a <= b & c <= d holds NE-corner (rectangle (a,b,c,d)) = |[b,d]| proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies NE-corner (rectangle (a,b,c,d)) = |[b,d]| ) set K = rectangle (a,b,c,d); assume that A1: a <= b and A2: c <= d ; ::_thesis: NE-corner (rectangle (a,b,c,d)) = |[b,d]| A3: NE-corner (rectangle (a,b,c,d)) = |[(E-bound (rectangle (a,b,c,d))),(N-bound (rectangle (a,b,c,d)))]| by PSCOMP_1:def_13; E-bound (rectangle (a,b,c,d)) = b by A1, A2, Th38; hence NE-corner (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, A3, Th37; ::_thesis: verum end; theorem :: JGRAPH_6:42 for a, b, c, d being real number st a <= b & c <= d holds SW-corner (rectangle (a,b,c,d)) = |[a,c]| proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies SW-corner (rectangle (a,b,c,d)) = |[a,c]| ) set K = rectangle (a,b,c,d); assume that A1: a <= b and A2: c <= d ; ::_thesis: SW-corner (rectangle (a,b,c,d)) = |[a,c]| A3: SW-corner (rectangle (a,b,c,d)) = |[(W-bound (rectangle (a,b,c,d))),(S-bound (rectangle (a,b,c,d)))]| by PSCOMP_1:def_11; W-bound (rectangle (a,b,c,d)) = a by A1, A2, Th36; hence SW-corner (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, A3, Th39; ::_thesis: verum end; theorem :: JGRAPH_6:43 for a, b, c, d being real number st a <= b & c <= d holds SE-corner (rectangle (a,b,c,d)) = |[b,c]| proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies SE-corner (rectangle (a,b,c,d)) = |[b,c]| ) set K = rectangle (a,b,c,d); assume that A1: a <= b and A2: c <= d ; ::_thesis: SE-corner (rectangle (a,b,c,d)) = |[b,c]| A3: SE-corner (rectangle (a,b,c,d)) = |[(E-bound (rectangle (a,b,c,d))),(S-bound (rectangle (a,b,c,d)))]| by PSCOMP_1:def_14; E-bound (rectangle (a,b,c,d)) = b by A1, A2, Th38; hence SE-corner (rectangle (a,b,c,d)) = |[b,c]| by A1, A2, A3, Th39; ::_thesis: verum end; theorem Th44: :: JGRAPH_6:44 for a, b, c, d being real number st a <= b & c <= d holds W-most (rectangle (a,b,c,d)) = LSeg (|[a,c]|,|[a,d]|) proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies W-most (rectangle (a,b,c,d)) = LSeg (|[a,c]|,|[a,d]|) ) set K = rectangle (a,b,c,d); assume that A1: a <= b and A2: c <= d ; ::_thesis: W-most (rectangle (a,b,c,d)) = LSeg (|[a,c]|,|[a,d]|) rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3; then A3: (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) c= rectangle (a,b,c,d) by XBOOLE_1:7; A4: LSeg (|[a,c]|,|[a,d]|) c= (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by XBOOLE_1:7; A5: SW-corner (rectangle (a,b,c,d)) = |[(W-bound (rectangle (a,b,c,d))),(S-bound (rectangle (a,b,c,d)))]| by PSCOMP_1:def_11; A6: NW-corner (rectangle (a,b,c,d)) = |[a,d]| by A1, A2, Th40; A7: W-bound (rectangle (a,b,c,d)) = a by A1, A2, Th36; A8: S-bound (rectangle (a,b,c,d)) = c by A1, A2, Th39; thus W-most (rectangle (a,b,c,d)) = (LSeg ((SW-corner (rectangle (a,b,c,d))),(NW-corner (rectangle (a,b,c,d))))) /\ (rectangle (a,b,c,d)) by PSCOMP_1:def_15 .= LSeg (|[a,c]|,|[a,d]|) by A3, A4, A5, A6, A7, A8, XBOOLE_1:1, XBOOLE_1:28 ; ::_thesis: verum end; theorem Th45: :: JGRAPH_6:45 for a, b, c, d being real number st a <= b & c <= d holds E-most (rectangle (a,b,c,d)) = LSeg (|[b,c]|,|[b,d]|) proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies E-most (rectangle (a,b,c,d)) = LSeg (|[b,c]|,|[b,d]|) ) set K = rectangle (a,b,c,d); assume that A1: a <= b and A2: c <= d ; ::_thesis: E-most (rectangle (a,b,c,d)) = LSeg (|[b,c]|,|[b,d]|) rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3; then A3: (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) c= rectangle (a,b,c,d) by XBOOLE_1:7; A4: LSeg (|[b,c]|,|[b,d]|) c= (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by XBOOLE_1:7; A5: SE-corner (rectangle (a,b,c,d)) = |[(E-bound (rectangle (a,b,c,d))),(S-bound (rectangle (a,b,c,d)))]| by PSCOMP_1:def_14; A6: NE-corner (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th41; A7: E-bound (rectangle (a,b,c,d)) = b by A1, A2, Th38; A8: S-bound (rectangle (a,b,c,d)) = c by A1, A2, Th39; thus E-most (rectangle (a,b,c,d)) = (LSeg ((SE-corner (rectangle (a,b,c,d))),(NE-corner (rectangle (a,b,c,d))))) /\ (rectangle (a,b,c,d)) by PSCOMP_1:def_17 .= LSeg (|[b,c]|,|[b,d]|) by A3, A4, A5, A6, A7, A8, XBOOLE_1:1, XBOOLE_1:28 ; ::_thesis: verum end; theorem Th46: :: JGRAPH_6:46 for a, b, c, d being real number st a <= b & c <= d holds ( W-min (rectangle (a,b,c,d)) = |[a,c]| & E-max (rectangle (a,b,c,d)) = |[b,d]| ) proof let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies ( W-min (rectangle (a,b,c,d)) = |[a,c]| & E-max (rectangle (a,b,c,d)) = |[b,d]| ) ) set K = rectangle (a,b,c,d); assume that A1: a <= b and A2: c <= d ; ::_thesis: ( W-min (rectangle (a,b,c,d)) = |[a,c]| & E-max (rectangle (a,b,c,d)) = |[b,d]| ) A3: lower_bound (proj2 | (LSeg (|[a,c]|,|[a,d]|))) = c proof set X = LSeg (|[a,c]|,|[a,d]|); reconsider Z = (proj2 | (LSeg (|[a,c]|,|[a,d]|))) .: the carrier of ((TOP-REAL 2) | (LSeg (|[a,c]|,|[a,d]|))) as Subset of REAL ; A4: LSeg (|[a,c]|,|[a,d]|) = the carrier of ((TOP-REAL 2) | (LSeg (|[a,c]|,|[a,d]|))) by PRE_TOPC:8; A5: for p being real number st p in Z holds p >= c proof let p be real number ; ::_thesis: ( p in Z implies p >= c ) assume p in Z ; ::_thesis: p >= c then consider p0 being set such that A6: p0 in the carrier of ((TOP-REAL 2) | (LSeg (|[a,c]|,|[a,d]|))) and p0 in the carrier of ((TOP-REAL 2) | (LSeg (|[a,c]|,|[a,d]|))) and A7: p = (proj2 | (LSeg (|[a,c]|,|[a,d]|))) . p0 by FUNCT_2:64; reconsider p0 = p0 as Point of (TOP-REAL 2) by A4, A6; A8: |[a,c]| `2 = c by EUCLID:52; |[a,d]| `2 = d by EUCLID:52; then p0 `2 >= c by A2, A4, A6, A8, TOPREAL1:4; hence p >= c by A4, A6, A7, PSCOMP_1:23; ::_thesis: verum end; A9: for q being real number st ( for p being real number st p in Z holds p >= q ) holds c >= q proof let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds p >= q ) implies c >= q ) assume A10: for p being real number st p in Z holds p >= q ; ::_thesis: c >= q A11: |[a,c]| in LSeg (|[a,c]|,|[a,d]|) by RLTOPSP1:68; (proj2 | (LSeg (|[a,c]|,|[a,d]|))) . |[a,c]| = |[a,c]| `2 by PSCOMP_1:23, RLTOPSP1:68 .= c by EUCLID:52 ; hence c >= q by A4, A10, A11, FUNCT_2:35; ::_thesis: verum end; thus lower_bound (proj2 | (LSeg (|[a,c]|,|[a,d]|))) = lower_bound Z by PSCOMP_1:def_1 .= c by A5, A9, SEQ_4:44 ; ::_thesis: verum end; A12: W-most (rectangle (a,b,c,d)) = LSeg (|[a,c]|,|[a,d]|) by A1, A2, Th44; A13: W-bound (rectangle (a,b,c,d)) = a by A1, A2, Th36; A14: upper_bound (proj2 | (LSeg (|[b,c]|,|[b,d]|))) = d proof set X = LSeg (|[b,c]|,|[b,d]|); reconsider Z = (proj2 | (LSeg (|[b,c]|,|[b,d]|))) .: the carrier of ((TOP-REAL 2) | (LSeg (|[b,c]|,|[b,d]|))) as Subset of REAL ; A15: LSeg (|[b,c]|,|[b,d]|) = the carrier of ((TOP-REAL 2) | (LSeg (|[b,c]|,|[b,d]|))) by PRE_TOPC:8; A16: for p being real number st p in Z holds p <= d proof let p be real number ; ::_thesis: ( p in Z implies p <= d ) assume p in Z ; ::_thesis: p <= d then consider p0 being set such that A17: p0 in the carrier of ((TOP-REAL 2) | (LSeg (|[b,c]|,|[b,d]|))) and p0 in the carrier of ((TOP-REAL 2) | (LSeg (|[b,c]|,|[b,d]|))) and A18: p = (proj2 | (LSeg (|[b,c]|,|[b,d]|))) . p0 by FUNCT_2:64; reconsider p0 = p0 as Point of (TOP-REAL 2) by A15, A17; A19: |[b,c]| `2 = c by EUCLID:52; |[b,d]| `2 = d by EUCLID:52; then p0 `2 <= d by A2, A15, A17, A19, TOPREAL1:4; hence p <= d by A15, A17, A18, PSCOMP_1:23; ::_thesis: verum end; A20: for q being real number st ( for p being real number st p in Z holds p <= q ) holds d <= q proof let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds p <= q ) implies d <= q ) assume A21: for p being real number st p in Z holds p <= q ; ::_thesis: d <= q A22: |[b,d]| in LSeg (|[b,c]|,|[b,d]|) by RLTOPSP1:68; (proj2 | (LSeg (|[b,c]|,|[b,d]|))) . |[b,d]| = |[b,d]| `2 by PSCOMP_1:23, RLTOPSP1:68 .= d by EUCLID:52 ; hence d <= q by A15, A21, A22, FUNCT_2:35; ::_thesis: verum end; thus upper_bound (proj2 | (LSeg (|[b,c]|,|[b,d]|))) = upper_bound Z by PSCOMP_1:def_2 .= d by A16, A20, SEQ_4:46 ; ::_thesis: verum end; A23: E-most (rectangle (a,b,c,d)) = LSeg (|[b,c]|,|[b,d]|) by A1, A2, Th45; E-bound (rectangle (a,b,c,d)) = b by A1, A2, Th38; hence ( W-min (rectangle (a,b,c,d)) = |[a,c]| & E-max (rectangle (a,b,c,d)) = |[b,d]| ) by A3, A12, A13, A14, A23, PSCOMP_1:def_19, PSCOMP_1:def_23; ::_thesis: verum end; theorem Th47: :: JGRAPH_6:47 for a, b, c, d being real number st a < b & c < d holds ( (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) is_an_arc_of W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) & (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) is_an_arc_of E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) ) proof let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies ( (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) is_an_arc_of W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) & (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) is_an_arc_of E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d ; ::_thesis: ( (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) is_an_arc_of W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) & (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) is_an_arc_of E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) ) A3: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; A4: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; |[a,c]| `2 = c by EUCLID:52; then A5: |[a,c]| <> |[a,d]| by A2, EUCLID:52; set p1 = |[a,c]|; set p2 = |[a,d]|; set q1 = |[b,d]|; A6: (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) = {|[a,d]|} by A1, A2, Th34; |[a,c]| `1 = a by EUCLID:52; then A7: |[a,c]| <> |[b,c]| by A1, EUCLID:52; set q2 = |[b,c]|; (LSeg (|[b,d]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[a,c]|)) = {|[b,c]|} by A1, A2, Th32; hence ( (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) is_an_arc_of W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) & (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) is_an_arc_of E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) ) by A3, A4, A5, A6, A7, TOPREAL1:12; ::_thesis: verum end; theorem Th48: :: JGRAPH_6:48 for a, b, c, d being real number for f1, f2 being FinSequence of (TOP-REAL 2) for p0, p1, p01, p10 being Point of (TOP-REAL 2) st a < b & c < d & p0 = |[a,c]| & p1 = |[b,d]| & p01 = |[a,d]| & p10 = |[b,c]| & f1 = <*p0,p01,p1*> & f2 = <*p0,p10,p1*> holds ( f1 is being_S-Seq & L~ f1 = (LSeg (p0,p01)) \/ (LSeg (p01,p1)) & f2 is being_S-Seq & L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) proof let a, b, c, d be real number ; ::_thesis: for f1, f2 being FinSequence of (TOP-REAL 2) for p0, p1, p01, p10 being Point of (TOP-REAL 2) st a < b & c < d & p0 = |[a,c]| & p1 = |[b,d]| & p01 = |[a,d]| & p10 = |[b,c]| & f1 = <*p0,p01,p1*> & f2 = <*p0,p10,p1*> holds ( f1 is being_S-Seq & L~ f1 = (LSeg (p0,p01)) \/ (LSeg (p01,p1)) & f2 is being_S-Seq & L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) let f1, f2 be FinSequence of (TOP-REAL 2); ::_thesis: for p0, p1, p01, p10 being Point of (TOP-REAL 2) st a < b & c < d & p0 = |[a,c]| & p1 = |[b,d]| & p01 = |[a,d]| & p10 = |[b,c]| & f1 = <*p0,p01,p1*> & f2 = <*p0,p10,p1*> holds ( f1 is being_S-Seq & L~ f1 = (LSeg (p0,p01)) \/ (LSeg (p01,p1)) & f2 is being_S-Seq & L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) let p0, p1, p01, p10 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p0 = |[a,c]| & p1 = |[b,d]| & p01 = |[a,d]| & p10 = |[b,c]| & f1 = <*p0,p01,p1*> & f2 = <*p0,p10,p1*> implies ( f1 is being_S-Seq & L~ f1 = (LSeg (p0,p01)) \/ (LSeg (p01,p1)) & f2 is being_S-Seq & L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) ) assume that A1: a < b and A2: c < d and A3: p0 = |[a,c]| and A4: p1 = |[b,d]| and A5: p01 = |[a,d]| and A6: p10 = |[b,c]| and A7: f1 = <*p0,p01,p1*> and A8: f2 = <*p0,p10,p1*> ; ::_thesis: ( f1 is being_S-Seq & L~ f1 = (LSeg (p0,p01)) \/ (LSeg (p01,p1)) & f2 is being_S-Seq & L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) set P = rectangle (a,b,c,d); set L1 = { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } ; set L2 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } ; set L3 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } ; set L4 = { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } ; A9: p1 `1 = b by A4, EUCLID:52; A10: p1 `2 = d by A4, EUCLID:52; A11: p10 `1 = b by A6, EUCLID:52; A12: p10 `2 = c by A6, EUCLID:52; A13: p0 `1 = a by A3, EUCLID:52; A14: p0 `2 = c by A3, EUCLID:52; A15: len f1 = 1 + 2 by A7, FINSEQ_1:45; A16: f1 /. 1 = p0 by A7, FINSEQ_4:18; A17: f1 /. 2 = p01 by A7, FINSEQ_4:18; A18: f1 /. 3 = p1 by A7, FINSEQ_4:18; thus f1 is being_S-Seq ::_thesis: ( L~ f1 = (LSeg (p0,p01)) \/ (LSeg (p01,p1)) & f2 is being_S-Seq & L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) proof A19: p0 <> p01 by A2, A5, A14, EUCLID:52; p01 <> p1 by A1, A5, A9, EUCLID:52; hence f1 is one-to-one by A1, A7, A9, A13, A19, FINSEQ_3:95; :: according to TOPREAL1:def_8 ::_thesis: ( 2 <= len f1 & f1 is unfolded & f1 is s.n.c. & f1 is special ) thus len f1 >= 2 by A15; ::_thesis: ( f1 is unfolded & f1 is s.n.c. & f1 is special ) thus f1 is unfolded ::_thesis: ( f1 is s.n.c. & f1 is special ) proof let i be Nat; :: according to TOPREAL1:def_6 ::_thesis: ( not 1 <= i or not i + 2 <= len f1 or (LSeg (f1,i)) /\ (LSeg (f1,(i + 1))) = {(f1 /. (i + 1))} ) assume that A20: 1 <= i and A21: i + 2 <= len f1 ; ::_thesis: (LSeg (f1,i)) /\ (LSeg (f1,(i + 1))) = {(f1 /. (i + 1))} i <= 1 by A15, A21, XREAL_1:6; then A22: i = 1 by A20, XXREAL_0:1; reconsider n2 = 1 + 1 as Element of NAT ; n2 in Seg (len f1) by A15, FINSEQ_1:1; then A23: LSeg (f1,1) = LSeg (p0,p01) by A15, A16, A17, TOPREAL1:def_3; A24: LSeg (f1,n2) = LSeg (p01,p1) by A15, A17, A18, TOPREAL1:def_3; for x being set holds ( x in (LSeg (p0,p01)) /\ (LSeg (p01,p1)) iff x = p01 ) proof let x be set ; ::_thesis: ( x in (LSeg (p0,p01)) /\ (LSeg (p01,p1)) iff x = p01 ) thus ( x in (LSeg (p0,p01)) /\ (LSeg (p01,p1)) implies x = p01 ) ::_thesis: ( x = p01 implies x in (LSeg (p0,p01)) /\ (LSeg (p01,p1)) ) proof assume A25: x in (LSeg (p0,p01)) /\ (LSeg (p01,p1)) ; ::_thesis: x = p01 then A26: x in LSeg (p0,p01) by XBOOLE_0:def_4; A27: x in LSeg (p01,p1) by A25, XBOOLE_0:def_4; A28: x in { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } by A2, A3, A5, A26, Th30; A29: x in { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) } by A1, A4, A5, A27, Th30; A30: ex p being Point of (TOP-REAL 2) st ( p = x & p `1 = a & p `2 <= d & p `2 >= c ) by A28; ex p2 being Point of (TOP-REAL 2) st ( p2 = x & p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) by A29; hence x = p01 by A5, A30, EUCLID:53; ::_thesis: verum end; assume A31: x = p01 ; ::_thesis: x in (LSeg (p0,p01)) /\ (LSeg (p01,p1)) then A32: x in LSeg (p0,p01) by RLTOPSP1:68; x in LSeg (p01,p1) by A31, RLTOPSP1:68; hence x in (LSeg (p0,p01)) /\ (LSeg (p01,p1)) by A32, XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg (f1,i)) /\ (LSeg (f1,(i + 1))) = {(f1 /. (i + 1))} by A17, A22, A23, A24, TARSKI:def_1; ::_thesis: verum end; thus f1 is s.n.c. ::_thesis: f1 is special proof let i, j be Nat; :: according to TOPREAL1:def_7 ::_thesis: ( j <= i + 1 or LSeg (f1,i) misses LSeg (f1,j) ) assume A33: i + 1 < j ; ::_thesis: LSeg (f1,i) misses LSeg (f1,j) now__::_thesis:_(LSeg_(f1,i))_/\_(LSeg_(f1,j))_=_{} percases ( 1 <= i or not 1 <= i or not i + 1 <= len f1 ) ; suppose 1 <= i ; ::_thesis: (LSeg (f1,i)) /\ (LSeg (f1,j)) = {} then A34: 1 + 1 <= i + 1 by XREAL_1:6; now__::_thesis:_(_(_1_<=_j_&_j_+_1_<=_len_f1_&_contradiction_)_or_(_(_not_1_<=_j_or_not_j_+_1_<=_len_f1_)_&_(LSeg_(f1,i))_/\_(LSeg_(f1,j))_=_{}_)_) percases ( ( 1 <= j & j + 1 <= len f1 ) or not 1 <= j or not j + 1 <= len f1 ) ; case ( 1 <= j & j + 1 <= len f1 ) ; ::_thesis: contradiction then j <= 2 by A15, XREAL_1:6; hence contradiction by A33, A34, XXREAL_0:2; ::_thesis: verum end; case ( not 1 <= j or not j + 1 <= len f1 ) ; ::_thesis: (LSeg (f1,i)) /\ (LSeg (f1,j)) = {} then LSeg (f1,j) = {} by TOPREAL1:def_3; hence (LSeg (f1,i)) /\ (LSeg (f1,j)) = {} ; ::_thesis: verum end; end; end; hence (LSeg (f1,i)) /\ (LSeg (f1,j)) = {} ; ::_thesis: verum end; suppose ( not 1 <= i or not i + 1 <= len f1 ) ; ::_thesis: (LSeg (f1,i)) /\ (LSeg (f1,j)) = {} then LSeg (f1,i) = {} by TOPREAL1:def_3; hence (LSeg (f1,i)) /\ (LSeg (f1,j)) = {} ; ::_thesis: verum end; end; end; hence (LSeg (f1,i)) /\ (LSeg (f1,j)) = {} ; :: according to XBOOLE_0:def_7 ::_thesis: verum end; let i be Nat; :: according to TOPREAL1:def_5 ::_thesis: ( not 1 <= i or not i + 1 <= len f1 or (f1 /. i) `1 = (f1 /. (i + 1)) `1 or (f1 /. i) `2 = (f1 /. (i + 1)) `2 ) assume that A35: 1 <= i and A36: i + 1 <= len f1 ; ::_thesis: ( (f1 /. i) `1 = (f1 /. (i + 1)) `1 or (f1 /. i) `2 = (f1 /. (i + 1)) `2 ) A37: i <= 1 + 1 by A15, A36, XREAL_1:6; now__::_thesis:_(_(f1_/._i)_`1_=_(f1_/._(i_+_1))_`1_or_(f1_/._i)_`2_=_(f1_/._(i_+_1))_`2_) percases ( i = 1 or i = 2 ) by A35, A37, NAT_1:9; supposeA38: i = 1 ; ::_thesis: ( (f1 /. i) `1 = (f1 /. (i + 1)) `1 or (f1 /. i) `2 = (f1 /. (i + 1)) `2 ) then (f1 /. i) `1 = p0 `1 by A7, FINSEQ_4:18 .= a by A3, EUCLID:52 .= (f1 /. (i + 1)) `1 by A5, A17, A38, EUCLID:52 ; hence ( (f1 /. i) `1 = (f1 /. (i + 1)) `1 or (f1 /. i) `2 = (f1 /. (i + 1)) `2 ) ; ::_thesis: verum end; supposeA39: i = 2 ; ::_thesis: ( (f1 /. i) `1 = (f1 /. (i + 1)) `1 or (f1 /. i) `2 = (f1 /. (i + 1)) `2 ) then (f1 /. i) `2 = p01 `2 by A7, FINSEQ_4:18 .= d by A5, EUCLID:52 .= (f1 /. (i + 1)) `2 by A4, A18, A39, EUCLID:52 ; hence ( (f1 /. i) `1 = (f1 /. (i + 1)) `1 or (f1 /. i) `2 = (f1 /. (i + 1)) `2 ) ; ::_thesis: verum end; end; end; hence ( (f1 /. i) `1 = (f1 /. (i + 1)) `1 or (f1 /. i) `2 = (f1 /. (i + 1)) `2 ) ; ::_thesis: verum end; A40: 1 + 1 in Seg (len f1) by A15, FINSEQ_1:1; A41: 1 + 1 <= len f1 by A15; LSeg (p0,p01) = LSeg (f1,1) by A15, A16, A17, A40, TOPREAL1:def_3; then A42: LSeg (p0,p01) in { (LSeg (f1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f1 ) } by A41; LSeg (p01,p1) = LSeg (f1,2) by A15, A17, A18, TOPREAL1:def_3; then LSeg (p01,p1) in { (LSeg (f1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f1 ) } by A15; then A43: {(LSeg (p0,p01)),(LSeg (p01,p1))} c= { (LSeg (f1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f1 ) } by A42, ZFMISC_1:32; { (LSeg (f1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f1 ) } c= {(LSeg (p0,p01)),(LSeg (p01,p1))} proof let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { (LSeg (f1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f1 ) } or a in {(LSeg (p0,p01)),(LSeg (p01,p1))} ) assume a in { (LSeg (f1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f1 ) } ; ::_thesis: a in {(LSeg (p0,p01)),(LSeg (p01,p1))} then consider i being Element of NAT such that A44: a = LSeg (f1,i) and A45: 1 <= i and A46: i + 1 <= len f1 ; i + 1 <= 2 + 1 by A7, A46, FINSEQ_1:45; then i <= 1 + 1 by XREAL_1:6; then ( i = 1 or i = 2 ) by A45, NAT_1:9; then ( a = LSeg (p0,p01) or a = LSeg (p01,p1) ) by A16, A17, A18, A44, A46, TOPREAL1:def_3; hence a in {(LSeg (p0,p01)),(LSeg (p01,p1))} by TARSKI:def_2; ::_thesis: verum end; then L~ f1 = union {(LSeg (p0,p01)),(LSeg (p01,p1))} by A43, XBOOLE_0:def_10; hence A47: L~ f1 = (LSeg (p0,p01)) \/ (LSeg (p01,p1)) by ZFMISC_1:75; ::_thesis: ( f2 is being_S-Seq & L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) then A48: L~ f1 = { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } \/ (LSeg (p01,p1)) by A2, A3, A5, Th30 .= { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } \/ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } by A1, A4, A5, Th30 ; A49: len f2 = 1 + 2 by A8, FINSEQ_1:45; A50: f2 /. 1 = p0 by A8, FINSEQ_4:18; A51: f2 /. 2 = p10 by A8, FINSEQ_4:18; A52: f2 /. 3 = p1 by A8, FINSEQ_4:18; thus f2 is being_S-Seq ::_thesis: ( L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) proof thus f2 is one-to-one by A1, A2, A8, A9, A10, A11, A12, A13, FINSEQ_3:95; :: according to TOPREAL1:def_8 ::_thesis: ( 2 <= len f2 & f2 is unfolded & f2 is s.n.c. & f2 is special ) thus len f2 >= 2 by A49; ::_thesis: ( f2 is unfolded & f2 is s.n.c. & f2 is special ) thus f2 is unfolded ::_thesis: ( f2 is s.n.c. & f2 is special ) proof let i be Nat; :: according to TOPREAL1:def_6 ::_thesis: ( not 1 <= i or not i + 2 <= len f2 or (LSeg (f2,i)) /\ (LSeg (f2,(i + 1))) = {(f2 /. (i + 1))} ) assume that A53: 1 <= i and A54: i + 2 <= len f2 ; ::_thesis: (LSeg (f2,i)) /\ (LSeg (f2,(i + 1))) = {(f2 /. (i + 1))} i <= 1 by A49, A54, XREAL_1:6; then A55: i = 1 by A53, XXREAL_0:1; 1 + 1 in Seg (len f2) by A49, FINSEQ_1:1; then A56: LSeg (f2,1) = LSeg (p0,p10) by A49, A50, A51, TOPREAL1:def_3; A57: LSeg (f2,(1 + 1)) = LSeg (p10,p1) by A49, A51, A52, TOPREAL1:def_3; for x being set holds ( x in (LSeg (p0,p10)) /\ (LSeg (p10,p1)) iff x = p10 ) proof let x be set ; ::_thesis: ( x in (LSeg (p0,p10)) /\ (LSeg (p10,p1)) iff x = p10 ) thus ( x in (LSeg (p0,p10)) /\ (LSeg (p10,p1)) implies x = p10 ) ::_thesis: ( x = p10 implies x in (LSeg (p0,p10)) /\ (LSeg (p10,p1)) ) proof assume A58: x in (LSeg (p0,p10)) /\ (LSeg (p10,p1)) ; ::_thesis: x = p10 then A59: x in LSeg (p0,p10) by XBOOLE_0:def_4; A60: x in LSeg (p10,p1) by A58, XBOOLE_0:def_4; A61: x in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } by A1, A3, A6, A59, Th30; A62: x in { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = b & p2 `2 <= d & p2 `2 >= c ) } by A2, A4, A6, A60, Th30; A63: ex p being Point of (TOP-REAL 2) st ( p = x & p `1 <= b & p `1 >= a & p `2 = c ) by A61; ex p2 being Point of (TOP-REAL 2) st ( p2 = x & p2 `1 = b & p2 `2 <= d & p2 `2 >= c ) by A62; hence x = p10 by A6, A63, EUCLID:53; ::_thesis: verum end; assume A64: x = p10 ; ::_thesis: x in (LSeg (p0,p10)) /\ (LSeg (p10,p1)) then A65: x in LSeg (p0,p10) by RLTOPSP1:68; x in LSeg (p10,p1) by A64, RLTOPSP1:68; hence x in (LSeg (p0,p10)) /\ (LSeg (p10,p1)) by A65, XBOOLE_0:def_4; ::_thesis: verum end; hence (LSeg (f2,i)) /\ (LSeg (f2,(i + 1))) = {(f2 /. (i + 1))} by A51, A55, A56, A57, TARSKI:def_1; ::_thesis: verum end; thus f2 is s.n.c. ::_thesis: f2 is special proof let i, j be Nat; :: according to TOPREAL1:def_7 ::_thesis: ( j <= i + 1 or LSeg (f2,i) misses LSeg (f2,j) ) assume A66: i + 1 < j ; ::_thesis: LSeg (f2,i) misses LSeg (f2,j) now__::_thesis:_(LSeg_(f2,i))_/\_(LSeg_(f2,j))_=_{} percases ( 1 <= i or not 1 <= i or not i + 1 <= len f2 ) ; suppose 1 <= i ; ::_thesis: (LSeg (f2,i)) /\ (LSeg (f2,j)) = {} then A67: 1 + 1 <= i + 1 by XREAL_1:6; now__::_thesis:_(_(_1_<=_j_&_j_+_1_<=_len_f2_&_contradiction_)_or_(_(_not_1_<=_j_or_not_j_+_1_<=_len_f2_)_&_(LSeg_(f2,i))_/\_(LSeg_(f2,j))_=_{}_)_) percases ( ( 1 <= j & j + 1 <= len f2 ) or not 1 <= j or not j + 1 <= len f2 ) ; case ( 1 <= j & j + 1 <= len f2 ) ; ::_thesis: contradiction then j <= 2 by A49, XREAL_1:6; hence contradiction by A66, A67, XXREAL_0:2; ::_thesis: verum end; case ( not 1 <= j or not j + 1 <= len f2 ) ; ::_thesis: (LSeg (f2,i)) /\ (LSeg (f2,j)) = {} then LSeg (f2,j) = {} by TOPREAL1:def_3; hence (LSeg (f2,i)) /\ (LSeg (f2,j)) = {} ; ::_thesis: verum end; end; end; hence (LSeg (f2,i)) /\ (LSeg (f2,j)) = {} ; ::_thesis: verum end; suppose ( not 1 <= i or not i + 1 <= len f2 ) ; ::_thesis: (LSeg (f2,i)) /\ (LSeg (f2,j)) = {} then LSeg (f2,i) = {} by TOPREAL1:def_3; hence (LSeg (f2,i)) /\ (LSeg (f2,j)) = {} ; ::_thesis: verum end; end; end; hence (LSeg (f2,i)) /\ (LSeg (f2,j)) = {} ; :: according to XBOOLE_0:def_7 ::_thesis: verum end; let i be Nat; :: according to TOPREAL1:def_5 ::_thesis: ( not 1 <= i or not i + 1 <= len f2 or (f2 /. i) `1 = (f2 /. (i + 1)) `1 or (f2 /. i) `2 = (f2 /. (i + 1)) `2 ) assume that A68: 1 <= i and A69: i + 1 <= len f2 ; ::_thesis: ( (f2 /. i) `1 = (f2 /. (i + 1)) `1 or (f2 /. i) `2 = (f2 /. (i + 1)) `2 ) A70: i <= 1 + 1 by A49, A69, XREAL_1:6; percases ( i = 1 or i = 2 ) by A68, A70, NAT_1:9; supposeA71: i = 1 ; ::_thesis: ( (f2 /. i) `1 = (f2 /. (i + 1)) `1 or (f2 /. i) `2 = (f2 /. (i + 1)) `2 ) then (f2 /. i) `2 = p0 `2 by A8, FINSEQ_4:18 .= c by A3, EUCLID:52 .= (f2 /. (i + 1)) `2 by A6, A51, A71, EUCLID:52 ; hence ( (f2 /. i) `1 = (f2 /. (i + 1)) `1 or (f2 /. i) `2 = (f2 /. (i + 1)) `2 ) ; ::_thesis: verum end; supposeA72: i = 2 ; ::_thesis: ( (f2 /. i) `1 = (f2 /. (i + 1)) `1 or (f2 /. i) `2 = (f2 /. (i + 1)) `2 ) then (f2 /. i) `1 = p10 `1 by A8, FINSEQ_4:18 .= b by A6, EUCLID:52 .= (f2 /. (i + 1)) `1 by A4, A52, A72, EUCLID:52 ; hence ( (f2 /. i) `1 = (f2 /. (i + 1)) `1 or (f2 /. i) `2 = (f2 /. (i + 1)) `2 ) ; ::_thesis: verum end; end; end; A73: 1 + 1 in Seg (len f2) by A49, FINSEQ_1:1; A74: 1 + 1 <= len f2 by A49; LSeg (p0,p10) = LSeg (f2,1) by A49, A50, A51, A73, TOPREAL1:def_3; then A75: LSeg (p0,p10) in { (LSeg (f2,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f2 ) } by A74; LSeg (p10,p1) = LSeg (f2,2) by A49, A51, A52, TOPREAL1:def_3; then LSeg (p10,p1) in { (LSeg (f2,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f2 ) } by A49; then A76: {(LSeg (p0,p10)),(LSeg (p10,p1))} c= { (LSeg (f2,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f2 ) } by A75, ZFMISC_1:32; { (LSeg (f2,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f2 ) } c= {(LSeg (p0,p10)),(LSeg (p10,p1))} proof let ax be set ; :: according to TARSKI:def_3 ::_thesis: ( not ax in { (LSeg (f2,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f2 ) } or ax in {(LSeg (p0,p10)),(LSeg (p10,p1))} ) assume ax in { (LSeg (f2,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f2 ) } ; ::_thesis: ax in {(LSeg (p0,p10)),(LSeg (p10,p1))} then consider i being Element of NAT such that A77: ax = LSeg (f2,i) and A78: 1 <= i and A79: i + 1 <= len f2 ; i + 1 <= 2 + 1 by A8, A79, FINSEQ_1:45; then i <= 1 + 1 by XREAL_1:6; then ( i = 1 or i = 2 ) by A78, NAT_1:9; then ( ax = LSeg (p0,p10) or ax = LSeg (p10,p1) ) by A50, A51, A52, A77, A79, TOPREAL1:def_3; hence ax in {(LSeg (p0,p10)),(LSeg (p10,p1))} by TARSKI:def_2; ::_thesis: verum end; then A80: L~ f2 = union {(LSeg (p0,p10)),(LSeg (p10,p1))} by A76, XBOOLE_0:def_10; hence L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) by ZFMISC_1:75; ::_thesis: ( rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) L~ f2 = (LSeg (p0,p10)) \/ (LSeg (p10,p1)) by A80, ZFMISC_1:75; then A81: L~ f2 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } \/ (LSeg (p10,p1)) by A1, A3, A6, Th30 .= { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } \/ { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } by A2, A4, A6, Th30 ; rectangle (a,b,c,d) = ((LSeg (p0,p01)) \/ (LSeg (p01,p1))) \/ ((LSeg (p0,p10)) \/ (LSeg (p10,p1))) by A3, A4, A5, A6, SPPOL_2:def_3; hence rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) by A47, A80, ZFMISC_1:75; ::_thesis: ( (L~ f1) /\ (L~ f2) = {p0,p1} & f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) now__::_thesis:_not__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_p_`1_<=_b_&_p_`1_>=_a_&_p_`2_=_d_)__}__meets__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_p_`1_<=_b_&_p_`1_>=_a_&_p_`2_=_c_)__}_ assume { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } meets { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } ; ::_thesis: contradiction then consider x being set such that A82: x in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } and A83: x in { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } by XBOOLE_0:3; A84: ex p being Point of (TOP-REAL 2) st ( p = x & p `1 <= b & p `1 >= a & p `2 = d ) by A82; ex p2 being Point of (TOP-REAL 2) st ( p2 = x & p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) by A83; hence contradiction by A2, A84; ::_thesis: verum end; then A85: { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } /\ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } = {} by XBOOLE_0:def_7; A86: LSeg (|[a,c]|,|[a,d]|) = { p3 where p3 is Point of (TOP-REAL 2) : ( p3 `1 = a & p3 `2 <= d & p3 `2 >= c ) } by A2, Th30; A87: LSeg (|[a,d]|,|[b,d]|) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = d ) } by A1, Th30; A88: LSeg (|[a,c]|,|[b,c]|) = { q1 where q1 is Point of (TOP-REAL 2) : ( q1 `1 <= b & q1 `1 >= a & q1 `2 = c ) } by A1, Th30; A89: LSeg (|[b,c]|,|[b,d]|) = { q2 where q2 is Point of (TOP-REAL 2) : ( q2 `1 = b & q2 `2 <= d & q2 `2 >= c ) } by A2, Th30; A90: (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,c]|,|[b,c]|)) = {|[a,c]|} by A1, A2, Th31; A91: (LSeg (|[a,d]|,|[b,d]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,d]|} by A1, A2, Th33; now__::_thesis:_not__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_p_`1_=_a_&_p_`2_<=_d_&_p_`2_>=_c_)__}__meets__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_p_`1_=_b_&_p_`2_<=_d_&_p_`2_>=_c_)__}_ assume { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } meets { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } ; ::_thesis: contradiction then consider x being set such that A92: x in { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } and A93: x in { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } by XBOOLE_0:3; A94: ex p being Point of (TOP-REAL 2) st ( p = x & p `1 = a & p `2 <= d & p `2 >= c ) by A92; ex p2 being Point of (TOP-REAL 2) st ( p2 = x & p2 `1 = b & p2 `2 <= d & p2 `2 >= c ) by A93; hence contradiction by A1, A94; ::_thesis: verum end; then A95: { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } /\ { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } = {} by XBOOLE_0:def_7; thus (L~ f1) /\ (L~ f2) = (( { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } \/ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } ) /\ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } ) \/ (( { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } \/ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } ) /\ { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } ) by A48, A81, XBOOLE_1:23 .= (( { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } /\ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } ) \/ ( { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } /\ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } )) \/ (( { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } \/ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } ) /\ { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } ) by XBOOLE_1:23 .= ( { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } /\ { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = c ) } ) \/ (( { p where p is Point of (TOP-REAL 2) : ( p `1 = a & p `2 <= d & p `2 >= c ) } /\ { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } ) \/ ( { p where p is Point of (TOP-REAL 2) : ( p `1 <= b & p `1 >= a & p `2 = d ) } /\ { p where p is Point of (TOP-REAL 2) : ( p `1 = b & p `2 <= d & p `2 >= c ) } )) by A85, XBOOLE_1:23 .= {p0,p1} by A3, A4, A86, A87, A88, A89, A90, A91, A95, ENUMSET1:1 ; ::_thesis: ( f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) thus ( f1 /. 1 = p0 & f1 /. (len f1) = p1 & f2 /. 1 = p0 & f2 /. (len f2) = p1 ) by A7, A8, A15, A49, FINSEQ_4:18; ::_thesis: verum end; theorem Th49: :: JGRAPH_6:49 for P1, P2 being Subset of (TOP-REAL 2) for a, b, c, d being real number for f1, f2 being FinSequence of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 = |[a,c]| & p2 = |[b,d]| & f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> & f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> & P1 = L~ f1 & P2 = L~ f2 holds ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & not P1 is empty & not P2 is empty & rectangle (a,b,c,d) = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) proof let P1, P2 be Subset of (TOP-REAL 2); ::_thesis: for a, b, c, d being real number for f1, f2 being FinSequence of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 = |[a,c]| & p2 = |[b,d]| & f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> & f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> & P1 = L~ f1 & P2 = L~ f2 holds ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & not P1 is empty & not P2 is empty & rectangle (a,b,c,d) = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) let a, b, c, d be real number ; ::_thesis: for f1, f2 being FinSequence of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 = |[a,c]| & p2 = |[b,d]| & f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> & f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> & P1 = L~ f1 & P2 = L~ f2 holds ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & not P1 is empty & not P2 is empty & rectangle (a,b,c,d) = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) let f1, f2 be FinSequence of (TOP-REAL 2); ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 = |[a,c]| & p2 = |[b,d]| & f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> & f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> & P1 = L~ f1 & P2 = L~ f2 holds ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & not P1 is empty & not P2 is empty & rectangle (a,b,c,d) = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 = |[a,c]| & p2 = |[b,d]| & f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> & f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> & P1 = L~ f1 & P2 = L~ f2 implies ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & not P1 is empty & not P2 is empty & rectangle (a,b,c,d) = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ) assume that A1: a < b and A2: c < d and A3: p1 = |[a,c]| and A4: p2 = |[b,d]| and A5: f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> and A6: f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> and A7: P1 = L~ f1 and A8: P2 = L~ f2 ; ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & not P1 is empty & not P2 is empty & rectangle (a,b,c,d) = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) |[a,c]| `2 = c by EUCLID:52; then A9: ( |[a,c]| <> |[a,d]| or |[a,d]| <> |[b,d]| ) by A2, EUCLID:52; A10: P1 = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, A5, A6, A7, Th48; A11: (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) = {|[a,d]|} by A1, A2, Th34; |[b,c]| `2 = c by EUCLID:52; then A12: ( |[a,c]| <> |[b,c]| or |[b,c]| <> |[b,d]| ) by A2, EUCLID:52; A13: P2 = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by A1, A2, A5, A6, A8, Th48; (LSeg (|[a,c]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[b,d]|)) = {|[b,c]|} by A1, A2, Th32; hence ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & not P1 is empty & not P2 is empty & rectangle (a,b,c,d) = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, Th48, TOPREAL1:12; ::_thesis: verum end; theorem Th50: :: JGRAPH_6:50 for a, b, c, d being real number st a < b & c < d holds rectangle (a,b,c,d) is being_simple_closed_curve proof let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies rectangle (a,b,c,d) is being_simple_closed_curve ) assume that A1: a < b and A2: c < d ; ::_thesis: rectangle (a,b,c,d) is being_simple_closed_curve set P = rectangle (a,b,c,d); set p1 = |[a,c]|; set p2 = |[b,d]|; reconsider f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> as FinSequence of (TOP-REAL 2) ; reconsider f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> as FinSequence of (TOP-REAL 2) ; set P1 = L~ f1; set P2 = L~ f2; A3: ( a < b & c < d & rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) } & |[a,c]| = |[a,c]| & |[b,d]| = |[b,d]| & f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> & f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> & L~ f1 = L~ f1 & L~ f2 = L~ f2 implies ( L~ f1 is_an_arc_of |[a,c]|,|[b,d]| & L~ f2 is_an_arc_of |[a,c]|,|[b,d]| & not L~ f1 is empty & not L~ f2 is empty & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {|[a,c]|,|[b,d]|} ) ) by Th49; |[a,c]| `1 = a by EUCLID:52; then A4: |[a,c]| <> |[b,d]| by A1, EUCLID:52; |[a,c]| in (L~ f1) /\ (L~ f2) by A1, A2, A3, Lm15, TARSKI:def_2; then |[a,c]| in L~ f1 by XBOOLE_0:def_4; then A5: |[a,c]| in rectangle (a,b,c,d) by A1, A2, A3, Lm15, XBOOLE_0:def_3; |[b,d]| in (L~ f1) /\ (L~ f2) by A1, A2, A3, Lm15, TARSKI:def_2; then |[b,d]| in L~ f1 by XBOOLE_0:def_4; then |[b,d]| in rectangle (a,b,c,d) by A1, A2, A3, Lm15, XBOOLE_0:def_3; hence rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, A3, A4, A5, Lm15, TOPREAL2:6; ::_thesis: verum end; theorem Th51: :: JGRAPH_6:51 for a, b, c, d being real number st a < b & c < d holds Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) proof let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d ; ::_thesis: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) A3: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; set P = rectangle (a,b,c,d); A4: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; A5: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; reconsider U = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) as non empty Subset of (TOP-REAL 2) ; A6: U is_an_arc_of W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) by A1, A2, Th47; reconsider P3 = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) as non empty Subset of (TOP-REAL 2) ; A7: P3 is_an_arc_of E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A1, A2, Th47; reconsider f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*>, f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> as FinSequence of (TOP-REAL 2) ; set p0 = |[a,c]|; set p01 = |[a,d]|; set p10 = |[b,c]|; set p1 = |[b,d]|; A8: ( a < b & c < d & |[a,c]| = |[a,c]| & |[b,d]| = |[b,d]| & |[a,d]| = |[a,d]| & |[b,c]| = |[b,c]| & f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> & f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> implies ( f1 is being_S-Seq & L~ f1 = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) & f2 is being_S-Seq & L~ f2 = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {|[a,c]|,|[b,d]|} & f1 /. 1 = |[a,c]| & f1 /. (len f1) = |[b,d]| & f2 /. 1 = |[a,c]| & f2 /. (len f2) = |[b,d]| ) ) by Th48; A9: Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) = Vertical_Line ((a + (E-bound (rectangle (a,b,c,d)))) / 2) by A1, A2, Th36 .= Vertical_Line ((a + b) / 2) by A1, A2, Th38 ; set Q = Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2); reconsider a2 = a, b2 = b, c2 = c, d2 = d as Real by XREAL_0:def_1; A10: U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) = {|[((a + b) / 2),d]|} proof thus U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) c= {|[((a + b) / 2),d]|} :: according to XBOOLE_0:def_10 ::_thesis: {|[((a + b) / 2),d]|} c= U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) or x in {|[((a + b) / 2),d]|} ) assume A11: x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) ; ::_thesis: x in {|[((a + b) / 2),d]|} then A12: x in U by XBOOLE_0:def_4; x in Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by A11, XBOOLE_0:def_4; then x in { p where p is Point of (TOP-REAL 2) : p `1 = (a + b) / 2 } by A9, JORDAN6:def_6; then consider p being Point of (TOP-REAL 2) such that A13: x = p and A14: p `1 = (a + b) / 2 ; now__::_thesis:_not_p_in_LSeg_(|[a,c]|,|[a,d]|) assume p in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: contradiction then p `1 = a by TOPREAL3:11; hence contradiction by A1, A14; ::_thesis: verum end; then p in LSeg (|[a2,d2]|,|[b2,d2]|) by A12, A13, XBOOLE_0:def_3; then p `2 = d by TOPREAL3:12; then x = |[((a + b) / 2),d]| by A13, A14, EUCLID:53; hence x in {|[((a + b) / 2),d]|} by TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[((a + b) / 2),d]|} or x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) ) assume x in {|[((a + b) / 2),d]|} ; ::_thesis: x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) then A15: x = |[((a + b) / 2),d]| by TARSKI:def_1; |[((a + b) / 2),d]| `1 = (a + b) / 2 by EUCLID:52; then x in { p where p is Point of (TOP-REAL 2) : p `1 = (a + b) / 2 } by A15; then A16: x in Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by A9, JORDAN6:def_6; A17: |[b,d]| `1 = b by EUCLID:52; A18: |[b,d]| `2 = d by EUCLID:52; A19: |[a,d]| `1 = a by EUCLID:52; |[a,d]| `2 = d by EUCLID:52; then x in LSeg (|[b,d]|,|[a,d]|) by A1, A15, A17, A18, A19, TOPREAL3:13; then x in U by XBOOLE_0:def_3; hence x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; then |[((a + b) / 2),d]| in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) by TARSKI:def_1; then U meets Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by XBOOLE_0:4; then First_Point (U,(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2))) in {|[((a + b) / 2),d]|} by A6, A10, JORDAN5C:def_1; then First_Point (U,(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2))) = |[((a + b) / 2),d]| by TARSKI:def_1; then A20: (First_Point (U,(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)))) `2 = d by EUCLID:52; A21: P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) = {|[((a + b) / 2),c]|} proof thus P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) c= {|[((a + b) / 2),c]|} :: according to XBOOLE_0:def_10 ::_thesis: {|[((a + b) / 2),c]|} c= P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) or x in {|[((a + b) / 2),c]|} ) assume A22: x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) ; ::_thesis: x in {|[((a + b) / 2),c]|} then A23: x in P3 by XBOOLE_0:def_4; x in Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by A22, XBOOLE_0:def_4; then x in { p where p is Point of (TOP-REAL 2) : p `1 = (a + b) / 2 } by A9, JORDAN6:def_6; then consider p being Point of (TOP-REAL 2) such that A24: x = p and A25: p `1 = (a + b) / 2 ; now__::_thesis:_not_p_in_LSeg_(|[b,c]|,|[b,d]|) assume p in LSeg (|[b,c]|,|[b,d]|) ; ::_thesis: contradiction then p `1 = b by TOPREAL3:11; hence contradiction by A1, A25; ::_thesis: verum end; then p in LSeg (|[a2,c2]|,|[b2,c2]|) by A23, A24, XBOOLE_0:def_3; then p `2 = c by TOPREAL3:12; then x = |[((a + b) / 2),c]| by A24, A25, EUCLID:53; hence x in {|[((a + b) / 2),c]|} by TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[((a + b) / 2),c]|} or x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) ) assume x in {|[((a + b) / 2),c]|} ; ::_thesis: x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) then A26: x = |[((a + b) / 2),c]| by TARSKI:def_1; |[((a + b) / 2),c]| `1 = (a + b) / 2 by EUCLID:52; then x in { p where p is Point of (TOP-REAL 2) : p `1 = (a + b) / 2 } by A26; then A27: x in Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by A9, JORDAN6:def_6; A28: |[b,c]| `1 = b by EUCLID:52; A29: |[b,c]| `2 = c by EUCLID:52; A30: |[a,c]| `1 = a by EUCLID:52; |[a,c]| `2 = c by EUCLID:52; then |[((b + a) / 2),c]| in LSeg (|[a,c]|,|[b,c]|) by A1, A28, A29, A30, TOPREAL3:13; then x in P3 by A26, XBOOLE_0:def_3; hence x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) by A27, XBOOLE_0:def_4; ::_thesis: verum end; then |[((a + b) / 2),c]| in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) by TARSKI:def_1; then P3 meets Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by XBOOLE_0:4; then Last_Point (P3,(E-max (rectangle (a,b,c,d))),(W-min (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2))) in {|[((a + b) / 2),c]|} by A7, A21, JORDAN5C:def_2; then Last_Point (P3,(E-max (rectangle (a,b,c,d))),(W-min (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2))) = |[((a + b) / 2),c]| by TARSKI:def_1; then (Last_Point (P3,(E-max (rectangle (a,b,c,d))),(W-min (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)))) `2 = c by EUCLID:52; hence Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, A3, A4, A5, A6, A7, A8, A20, JORDAN6:def_8; ::_thesis: verum end; theorem Th52: :: JGRAPH_6:52 for a, b, c, d being real number st a < b & c < d holds Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) proof let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d ; ::_thesis: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) A3: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; set P = rectangle (a,b,c,d); A4: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; A5: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; reconsider U = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) as non empty Subset of (TOP-REAL 2) ; A6: U is_an_arc_of W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) by A1, A2, Th47; reconsider P3 = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) as non empty Subset of (TOP-REAL 2) ; A7: P3 is_an_arc_of E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A1, A2, Th47; reconsider f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*>, f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> as FinSequence of (TOP-REAL 2) ; set p0 = |[a,c]|; set p01 = |[a,d]|; set p10 = |[b,c]|; set p1 = |[b,d]|; A8: ( a < b & c < d & |[a,c]| = |[a,c]| & |[b,d]| = |[b,d]| & |[a,d]| = |[a,d]| & |[b,c]| = |[b,c]| & f1 = <*|[a,c]|,|[a,d]|,|[b,d]|*> & f2 = <*|[a,c]|,|[b,c]|,|[b,d]|*> implies ( f1 is being_S-Seq & L~ f1 = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) & f2 is being_S-Seq & L~ f2 = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) & rectangle (a,b,c,d) = (L~ f1) \/ (L~ f2) & (L~ f1) /\ (L~ f2) = {|[a,c]|,|[b,d]|} & f1 /. 1 = |[a,c]| & f1 /. (len f1) = |[b,d]| & f2 /. 1 = |[a,c]| & f2 /. (len f2) = |[b,d]| ) ) by Th48; A9: Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) = Vertical_Line ((a + (E-bound (rectangle (a,b,c,d)))) / 2) by A1, A2, Th36 .= Vertical_Line ((a + b) / 2) by A1, A2, Th38 ; set Q = Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2); reconsider a2 = a, b2 = b, c2 = c, d2 = d as Real by XREAL_0:def_1; A10: U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) = {|[((a + b) / 2),d]|} proof thus U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) c= {|[((a + b) / 2),d]|} :: according to XBOOLE_0:def_10 ::_thesis: {|[((a + b) / 2),d]|} c= U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) or x in {|[((a + b) / 2),d]|} ) assume A11: x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) ; ::_thesis: x in {|[((a + b) / 2),d]|} then A12: x in U by XBOOLE_0:def_4; x in Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by A11, XBOOLE_0:def_4; then x in { p where p is Point of (TOP-REAL 2) : p `1 = (a + b) / 2 } by A9, JORDAN6:def_6; then consider p being Point of (TOP-REAL 2) such that A13: x = p and A14: p `1 = (a + b) / 2 ; now__::_thesis:_not_p_in_LSeg_(|[a,c]|,|[a,d]|) assume p in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: contradiction then p `1 = a by TOPREAL3:11; hence contradiction by A1, A14; ::_thesis: verum end; then p in LSeg (|[a2,d2]|,|[b2,d2]|) by A12, A13, XBOOLE_0:def_3; then p `2 = d by TOPREAL3:12; then x = |[((a + b) / 2),d]| by A13, A14, EUCLID:53; hence x in {|[((a + b) / 2),d]|} by TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[((a + b) / 2),d]|} or x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) ) assume x in {|[((a + b) / 2),d]|} ; ::_thesis: x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) then A15: x = |[((a + b) / 2),d]| by TARSKI:def_1; |[((a + b) / 2),d]| `1 = (a + b) / 2 by EUCLID:52; then x in { p where p is Point of (TOP-REAL 2) : p `1 = (a + b) / 2 } by A15; then A16: x in Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by A9, JORDAN6:def_6; A17: |[b,d]| `1 = b by EUCLID:52; A18: |[b,d]| `2 = d by EUCLID:52; A19: |[a,d]| `1 = a by EUCLID:52; |[a,d]| `2 = d by EUCLID:52; then x in LSeg (|[b,d]|,|[a,d]|) by A1, A15, A17, A18, A19, TOPREAL3:13; then x in U by XBOOLE_0:def_3; hence x in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) by A16, XBOOLE_0:def_4; ::_thesis: verum end; then |[((a + b) / 2),d]| in U /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) by TARSKI:def_1; then U meets Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by XBOOLE_0:4; then First_Point (U,(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2))) in {|[((a + b) / 2),d]|} by A6, A10, JORDAN5C:def_1; then First_Point (U,(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2))) = |[((a + b) / 2),d]| by TARSKI:def_1; then A20: (First_Point (U,(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)))) `2 = d by EUCLID:52; A21: P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) = {|[((a + b) / 2),c]|} proof thus P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) c= {|[((a + b) / 2),c]|} :: according to XBOOLE_0:def_10 ::_thesis: {|[((a + b) / 2),c]|} c= P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) or x in {|[((a + b) / 2),c]|} ) assume A22: x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) ; ::_thesis: x in {|[((a + b) / 2),c]|} then A23: x in P3 by XBOOLE_0:def_4; x in Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by A22, XBOOLE_0:def_4; then x in { p where p is Point of (TOP-REAL 2) : p `1 = (a + b) / 2 } by A9, JORDAN6:def_6; then consider p being Point of (TOP-REAL 2) such that A24: x = p and A25: p `1 = (a + b) / 2 ; now__::_thesis:_not_p_in_LSeg_(|[b,c]|,|[b,d]|) assume p in LSeg (|[b,c]|,|[b,d]|) ; ::_thesis: contradiction then p `1 = b by TOPREAL3:11; hence contradiction by A1, A25; ::_thesis: verum end; then p in LSeg (|[a2,c2]|,|[b2,c2]|) by A23, A24, XBOOLE_0:def_3; then p `2 = c by TOPREAL3:12; then x = |[((a + b) / 2),c]| by A24, A25, EUCLID:53; hence x in {|[((a + b) / 2),c]|} by TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[((a + b) / 2),c]|} or x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) ) assume x in {|[((a + b) / 2),c]|} ; ::_thesis: x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) then A26: x = |[((a + b) / 2),c]| by TARSKI:def_1; |[((a + b) / 2),c]| `1 = (a + b) / 2 by EUCLID:52; then x in { p where p is Point of (TOP-REAL 2) : p `1 = (a + b) / 2 } by A26; then A27: x in Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by A9, JORDAN6:def_6; A28: |[b,c]| `1 = b by EUCLID:52; A29: |[b,c]| `2 = c by EUCLID:52; A30: |[a,c]| `1 = a by EUCLID:52; |[a,c]| `2 = c by EUCLID:52; then |[((a + b) / 2),c]| in LSeg (|[a,c]|,|[b,c]|) by A1, A28, A29, A30, TOPREAL3:13; then x in P3 by A26, XBOOLE_0:def_3; hence x in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) by A27, XBOOLE_0:def_4; ::_thesis: verum end; then |[((a + b) / 2),c]| in P3 /\ (Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)) by TARSKI:def_1; then P3 meets Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2) by XBOOLE_0:4; then Last_Point (P3,(E-max (rectangle (a,b,c,d))),(W-min (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2))) in {|[((a + b) / 2),c]|} by A7, A21, JORDAN5C:def_2; then Last_Point (P3,(E-max (rectangle (a,b,c,d))),(W-min (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2))) = |[((a + b) / 2),c]| by TARSKI:def_1; then A31: (Last_Point (P3,(E-max (rectangle (a,b,c,d))),(W-min (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)))) `2 = c by EUCLID:52; A32: P3 is_an_arc_of E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A1, A2, Th47; A33: (Upper_Arc (rectangle (a,b,c,d))) /\ P3 = {(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d)))} by A1, A2, A4, A5, A8, Th51; A34: (Upper_Arc (rectangle (a,b,c,d))) \/ P3 = rectangle (a,b,c,d) by A1, A2, A8, Th51; (First_Point ((Upper_Arc (rectangle (a,b,c,d))),(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)))) `2 > (Last_Point (P3,(E-max (rectangle (a,b,c,d))),(W-min (rectangle (a,b,c,d))),(Vertical_Line (((W-bound (rectangle (a,b,c,d))) + (E-bound (rectangle (a,b,c,d)))) / 2)))) `2 by A1, A2, A20, A31, Th51; hence Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by A3, A32, A33, A34, JORDAN6:def_9; ::_thesis: verum end; theorem Th53: :: JGRAPH_6:53 for a, b, c, d being real number st a < b & c < d holds ex f being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) st ( f is being_homeomorphism & f . 0 = W-min (rectangle (a,b,c,d)) & f . 1 = E-max (rectangle (a,b,c,d)) & rng f = Upper_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[a,c]|,|[a,d]|) holds ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f . ((((p `2) - c) / (d - c)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[a,d]|,|[b,d]|) holds ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) ) ) proof let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies ex f being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) st ( f is being_homeomorphism & f . 0 = W-min (rectangle (a,b,c,d)) & f . 1 = E-max (rectangle (a,b,c,d)) & rng f = Upper_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[a,c]|,|[a,d]|) holds ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f . ((((p `2) - c) / (d - c)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[a,d]|,|[b,d]|) holds ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) ) ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d ; ::_thesis: ex f being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) st ( f is being_homeomorphism & f . 0 = W-min (rectangle (a,b,c,d)) & f . 1 = E-max (rectangle (a,b,c,d)) & rng f = Upper_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[a,c]|,|[a,d]|) holds ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f . ((((p `2) - c) / (d - c)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[a,d]|,|[b,d]|) holds ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) ) ) defpred S1[ set , set ] means for r being Real st $1 = r holds ( ( r in [.0,(1 / 2).] implies $2 = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) ) & ( r in [.(1 / 2),1.] implies $2 = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) ) ); A3: [.0,1.] = [.0,(1 / 2).] \/ [.(1 / 2),1.] by XXREAL_1:165; A4: for x being set st x in [.0,1.] holds ex y being set st S1[x,y] proof let x be set ; ::_thesis: ( x in [.0,1.] implies ex y being set st S1[x,y] ) assume A5: x in [.0,1.] ; ::_thesis: ex y being set st S1[x,y] now__::_thesis:_(_(_x_in_[.0,(1_/_2).]_&_ex_y_being_set_st_S1[x,y]_)_or_(_x_in_[.(1_/_2),1.]_&_ex_y_being_set_st_S1[x,y]_)_) percases ( x in [.0,(1 / 2).] or x in [.(1 / 2),1.] ) by A3, A5, XBOOLE_0:def_3; caseA6: x in [.0,(1 / 2).] ; ::_thesis: ex y being set st S1[x,y] then reconsider r = x as Real ; A7: r <= 1 / 2 by A6, XXREAL_1:1; set y0 = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|); ( r in [.(1 / 2),1.] implies ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) ) proof assume r in [.(1 / 2),1.] ; ::_thesis: ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) then 1 / 2 <= r by XXREAL_1:1; then A8: r = 1 / 2 by A7, XXREAL_0:1; then A9: ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) = (0 * |[a,c]|) + |[a,d]| by EUCLID:29 .= (0. (TOP-REAL 2)) + |[a,d]| by EUCLID:29 .= |[a,d]| by EUCLID:27 ; ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) = (1 * |[a,d]|) + (0. (TOP-REAL 2)) by A8, EUCLID:29 .= |[a,d]| + (0. (TOP-REAL 2)) by EUCLID:29 .= |[a,d]| by EUCLID:27 ; hence ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) by A9; ::_thesis: verum end; then for r2 being Real st x = r2 holds ( ( r2 in [.0,(1 / 2).] implies ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) = ((1 - (2 * r2)) * |[a,c]|) + ((2 * r2) * |[a,d]|) ) & ( r2 in [.(1 / 2),1.] implies ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) = ((1 - ((2 * r2) - 1)) * |[a,d]|) + (((2 * r2) - 1) * |[b,d]|) ) ) ; hence ex y being set st S1[x,y] ; ::_thesis: verum end; caseA10: x in [.(1 / 2),1.] ; ::_thesis: ex y being set st S1[x,y] then reconsider r = x as Real ; A11: 1 / 2 <= r by A10, XXREAL_1:1; set y0 = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|); ( r in [.0,(1 / 2).] implies ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) ) proof assume r in [.0,(1 / 2).] ; ::_thesis: ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) then r <= 1 / 2 by XXREAL_1:1; then A12: r = 1 / 2 by A11, XXREAL_0:1; then A13: ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) = |[a,d]| + (0 * |[b,d]|) by EUCLID:29 .= |[a,d]| + (0. (TOP-REAL 2)) by EUCLID:29 .= |[a,d]| by EUCLID:27 ; ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) = (0. (TOP-REAL 2)) + (1 * |[a,d]|) by A12, EUCLID:29 .= (0. (TOP-REAL 2)) + |[a,d]| by EUCLID:29 .= |[a,d]| by EUCLID:27 ; hence ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) by A13; ::_thesis: verum end; then for r2 being Real st x = r2 holds ( ( r2 in [.0,(1 / 2).] implies ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) = ((1 - (2 * r2)) * |[a,c]|) + ((2 * r2) * |[a,d]|) ) & ( r2 in [.(1 / 2),1.] implies ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) = ((1 - ((2 * r2) - 1)) * |[a,d]|) + (((2 * r2) - 1) * |[b,d]|) ) ) ; hence ex y being set st S1[x,y] ; ::_thesis: verum end; end; end; hence ex y being set st S1[x,y] ; ::_thesis: verum end; ex f2 being Function st ( dom f2 = [.0,1.] & ( for x being set st x in [.0,1.] holds S1[x,f2 . x] ) ) from CLASSES1:sch_1(A4); then consider f2 being Function such that A14: dom f2 = [.0,1.] and A15: for x being set st x in [.0,1.] holds S1[x,f2 . x] ; rng f2 c= the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) ) assume y in rng f2 ; ::_thesis: y in the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) then consider x being set such that A16: x in dom f2 and A17: y = f2 . x by FUNCT_1:def_3; now__::_thesis:_(_(_x_in_[.0,(1_/_2).]_&_y_in_the_carrier_of_((TOP-REAL_2)_|_(Upper_Arc_(rectangle_(a,b,c,d))))_)_or_(_x_in_[.(1_/_2),1.]_&_y_in_the_carrier_of_((TOP-REAL_2)_|_(Upper_Arc_(rectangle_(a,b,c,d))))_)_) percases ( x in [.0,(1 / 2).] or x in [.(1 / 2),1.] ) by A3, A14, A16, XBOOLE_0:def_3; caseA18: x in [.0,(1 / 2).] ; ::_thesis: y in the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) then reconsider r = x as Real ; A19: 0 <= r by A18, XXREAL_1:1; r <= 1 / 2 by A18, XXREAL_1:1; then A20: r * 2 <= (1 / 2) * 2 by XREAL_1:64; f2 . x = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) by A14, A15, A16, A18; then A21: y in { (((1 - lambda) * |[a,c]|) + (lambda * |[a,d]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by A17, A19, A20; Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; then y in Upper_Arc (rectangle (a,b,c,d)) by A21, XBOOLE_0:def_3; hence y in the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) by PRE_TOPC:8; ::_thesis: verum end; caseA22: x in [.(1 / 2),1.] ; ::_thesis: y in the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) then reconsider r = x as Real ; A23: 1 / 2 <= r by A22, XXREAL_1:1; A24: r <= 1 by A22, XXREAL_1:1; r * 2 >= (1 / 2) * 2 by A23, XREAL_1:64; then A25: (2 * r) - 1 >= 0 by XREAL_1:48; 2 * 1 >= 2 * r by A24, XREAL_1:64; then A26: (1 + 1) - 1 >= (2 * r) - 1 by XREAL_1:9; f2 . x = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) by A14, A15, A16, A22; then A27: y in { (((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by A17, A25, A26; Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; then y in Upper_Arc (rectangle (a,b,c,d)) by A27, XBOOLE_0:def_3; hence y in the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) by PRE_TOPC:8; ::_thesis: verum end; end; end; hence y in the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) ; ::_thesis: verum end; then reconsider f3 = f2 as Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) by A14, BORSUK_1:40, FUNCT_2:2; A28: 0 in [.0,1.] by XXREAL_1:1; 0 in [.0,(1 / 2).] by XXREAL_1:1; then A29: f3 . 0 = ((1 - (2 * 0)) * |[a,c]|) + ((2 * 0) * |[a,d]|) by A15, A28 .= (1 * |[a,c]|) + (0. (TOP-REAL 2)) by EUCLID:29 .= |[a,c]| + (0. (TOP-REAL 2)) by EUCLID:29 .= |[a,c]| by EUCLID:27 .= W-min (rectangle (a,b,c,d)) by A1, A2, Th46 ; A30: 1 in [.0,1.] by XXREAL_1:1; 1 in [.(1 / 2),1.] by XXREAL_1:1; then A31: f3 . 1 = ((1 - ((2 * 1) - 1)) * |[a,d]|) + (((2 * 1) - 1) * |[b,d]|) by A15, A30 .= (0 * |[a,d]|) + |[b,d]| by EUCLID:29 .= (0. (TOP-REAL 2)) + |[b,d]| by EUCLID:29 .= |[b,d]| by EUCLID:27 .= E-max (rectangle (a,b,c,d)) by A1, A2, Th46 ; A32: for r being Real st r in [.0,(1 / 2).] holds f3 . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) proof let r be Real; ::_thesis: ( r in [.0,(1 / 2).] implies f3 . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) ) assume A33: r in [.0,(1 / 2).] ; ::_thesis: f3 . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) then A34: 0 <= r by XXREAL_1:1; r <= 1 / 2 by A33, XXREAL_1:1; then r <= 1 by XXREAL_0:2; then r in [.0,1.] by A34, XXREAL_1:1; hence f3 . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) by A15, A33; ::_thesis: verum end; A35: for r being Real st r in [.(1 / 2),1.] holds f3 . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) proof let r be Real; ::_thesis: ( r in [.(1 / 2),1.] implies f3 . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) ) assume A36: r in [.(1 / 2),1.] ; ::_thesis: f3 . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) then A37: 1 / 2 <= r by XXREAL_1:1; r <= 1 by A36, XXREAL_1:1; then r in [.0,1.] by A37, XXREAL_1:1; hence f3 . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) by A15, A36; ::_thesis: verum end; A38: for p being Point of (TOP-REAL 2) st p in LSeg (|[a,c]|,|[a,d]|) holds ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f3 . ((((p `2) - c) / (d - c)) / 2) = p ) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (|[a,c]|,|[a,d]|) implies ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f3 . ((((p `2) - c) / (d - c)) / 2) = p ) ) assume A39: p in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f3 . ((((p `2) - c) / (d - c)) / 2) = p ) A40: |[a,c]| `2 = c by EUCLID:52; A41: |[a,d]| `2 = d by EUCLID:52; then A42: c <= p `2 by A2, A39, A40, TOPREAL1:4; A43: p `2 <= d by A2, A39, A40, A41, TOPREAL1:4; A44: d - c > 0 by A2, XREAL_1:50; A45: (p `2) - c >= 0 by A42, XREAL_1:48; A46: d - c > 0 by A2, XREAL_1:50; (p `2) - c <= d - c by A43, XREAL_1:9; then ((p `2) - c) / (d - c) <= (d - c) / (d - c) by A46, XREAL_1:72; then ((p `2) - c) / (d - c) <= 1 by A46, XCMPLX_1:60; then A47: (((p `2) - c) / (d - c)) / 2 <= 1 / 2 by XREAL_1:72; set r = (((p `2) - c) / (d - c)) / 2; (((p `2) - c) / (d - c)) / 2 in [.0,(1 / 2).] by A44, A45, A47, XXREAL_1:1; then f3 . ((((p `2) - c) / (d - c)) / 2) = ((1 - (2 * ((((p `2) - c) / (d - c)) / 2))) * |[a,c]|) + ((2 * ((((p `2) - c) / (d - c)) / 2)) * |[a,d]|) by A32 .= |[((1 - (2 * ((((p `2) - c) / (d - c)) / 2))) * a),((1 - (2 * ((((p `2) - c) / (d - c)) / 2))) * c)]| + ((2 * ((((p `2) - c) / (d - c)) / 2)) * |[a,d]|) by EUCLID:58 .= |[((1 - (2 * ((((p `2) - c) / (d - c)) / 2))) * a),((1 - (2 * ((((p `2) - c) / (d - c)) / 2))) * c)]| + |[((2 * ((((p `2) - c) / (d - c)) / 2)) * a),((2 * ((((p `2) - c) / (d - c)) / 2)) * d)]| by EUCLID:58 .= |[(((1 * a) - ((2 * ((((p `2) - c) / (d - c)) / 2)) * a)) + ((2 * ((((p `2) - c) / (d - c)) / 2)) * a)),(((1 - (2 * ((((p `2) - c) / (d - c)) / 2))) * c) + ((2 * ((((p `2) - c) / (d - c)) / 2)) * d))]| by EUCLID:56 .= |[a,((1 * c) + ((((p `2) - c) / (d - c)) * (d - c)))]| .= |[a,((1 * c) + ((p `2) - c))]| by A46, XCMPLX_1:87 .= |[(p `1),(p `2)]| by A39, TOPREAL3:11 .= p by EUCLID:53 ; hence ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f3 . ((((p `2) - c) / (d - c)) / 2) = p ) by A44, A45, A47, XXREAL_0:2; ::_thesis: verum end; A48: for p being Point of (TOP-REAL 2) st p in LSeg (|[a,d]|,|[b,d]|) holds ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (|[a,d]|,|[b,d]|) implies ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) ) assume A49: p in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) A50: |[a,d]| `1 = a by EUCLID:52; A51: |[b,d]| `1 = b by EUCLID:52; then A52: a <= p `1 by A1, A49, A50, TOPREAL1:3; A53: p `1 <= b by A1, A49, A50, A51, TOPREAL1:3; A54: b - a > 0 by A1, XREAL_1:50; A55: (p `1) - a >= 0 by A52, XREAL_1:48; then A56: ((((p `1) - a) / (b - a)) / 2) + (1 / 2) >= 0 + (1 / 2) by A54, XREAL_1:7; A57: b - a > 0 by A1, XREAL_1:50; (p `1) - a <= b - a by A53, XREAL_1:9; then ((p `1) - a) / (b - a) <= (b - a) / (b - a) by A57, XREAL_1:72; then ((p `1) - a) / (b - a) <= 1 by A57, XCMPLX_1:60; then (((p `1) - a) / (b - a)) / 2 <= 1 / 2 by XREAL_1:72; then A58: ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= (1 / 2) + (1 / 2) by XREAL_1:7; set r = ((((p `1) - a) / (b - a)) / 2) + (1 / 2); ((((p `1) - a) / (b - a)) / 2) + (1 / 2) in [.(1 / 2),1.] by A56, A58, XXREAL_1:1; then f3 . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = ((1 - ((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1)) * |[a,d]|) + (((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1) * |[b,d]|) by A35 .= |[((1 - ((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1)) * a),((1 - ((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1)) * d)]| + (((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1) * |[b,d]|) by EUCLID:58 .= |[((1 - ((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1)) * a),((1 - ((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1)) * d)]| + |[(((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1) * b),(((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1) * d)]| by EUCLID:58 .= |[(((1 - ((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1)) * a) + (((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1) * b)),(((1 * d) - (((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1) * d)) + (((2 * (((((p `1) - a) / (b - a)) / 2) + (1 / 2))) - 1) * d))]| by EUCLID:56 .= |[((1 * a) + ((((p `1) - a) / (b - a)) * (b - a))),d]| .= |[((1 * a) + ((p `1) - a)),d]| by A57, XCMPLX_1:87 .= |[(p `1),(p `2)]| by A49, TOPREAL3:12 .= p by EUCLID:53 ; hence ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) by A54, A55, A58; ::_thesis: verum end; reconsider B00 = [.0,1.] as Subset of R^1 by TOPMETR:17; reconsider B01 = B00 as non empty Subset of R^1 by XXREAL_1:1; I[01] = R^1 | B01 by TOPMETR:19, TOPMETR:20; then consider h1 being Function of I[01],R^1 such that A59: for p being Point of I[01] holds h1 . p = p and A60: h1 is continuous by Th6; consider h2 being Function of I[01],R^1 such that A61: for p being Point of I[01] for r1 being real number st h1 . p = r1 holds h2 . p = 2 * r1 and A62: h2 is continuous by A60, JGRAPH_2:23; consider h5 being Function of I[01],R^1 such that A63: for p being Point of I[01] for r1 being real number st h2 . p = r1 holds h5 . p = 1 - r1 and A64: h5 is continuous by A62, Th8; consider h3 being Function of I[01],R^1 such that A65: for p being Point of I[01] for r1 being real number st h2 . p = r1 holds h3 . p = r1 - 1 and A66: h3 is continuous by A62, Th7; consider h4 being Function of I[01],R^1 such that A67: for p being Point of I[01] for r1 being real number st h3 . p = r1 holds h4 . p = 1 - r1 and A68: h4 is continuous by A66, Th8; consider g1 being Function of I[01],(TOP-REAL 2) such that A69: for r being Point of I[01] holds g1 . r = ((h5 . r) * |[a,c]|) + ((h2 . r) * |[a,d]|) and A70: g1 is continuous by A62, A64, Th13; A71: for r being Point of I[01] for s being real number st r = s holds g1 . r = ((1 - (2 * s)) * |[a,c]|) + ((2 * s) * |[a,d]|) proof let r be Point of I[01]; ::_thesis: for s being real number st r = s holds g1 . r = ((1 - (2 * s)) * |[a,c]|) + ((2 * s) * |[a,d]|) let s be real number ; ::_thesis: ( r = s implies g1 . r = ((1 - (2 * s)) * |[a,c]|) + ((2 * s) * |[a,d]|) ) assume A72: r = s ; ::_thesis: g1 . r = ((1 - (2 * s)) * |[a,c]|) + ((2 * s) * |[a,d]|) g1 . r = ((h5 . r) * |[a,c]|) + ((h2 . r) * |[a,d]|) by A69 .= ((1 - (2 * (h1 . r))) * |[a,c]|) + ((h2 . r) * |[a,d]|) by A61, A63 .= ((1 - (2 * (h1 . r))) * |[a,c]|) + ((2 * (h1 . r)) * |[a,d]|) by A61 .= ((1 - (2 * s)) * |[a,c]|) + ((2 * (h1 . r)) * |[a,d]|) by A59, A72 .= ((1 - (2 * s)) * |[a,c]|) + ((2 * s) * |[a,d]|) by A59, A72 ; hence g1 . r = ((1 - (2 * s)) * |[a,c]|) + ((2 * s) * |[a,d]|) ; ::_thesis: verum end; consider g2 being Function of I[01],(TOP-REAL 2) such that A73: for r being Point of I[01] holds g2 . r = ((h4 . r) * |[a,d]|) + ((h3 . r) * |[b,d]|) and A74: g2 is continuous by A66, A68, Th13; A75: for r being Point of I[01] for s being real number st r = s holds g2 . r = ((1 - ((2 * s) - 1)) * |[a,d]|) + (((2 * s) - 1) * |[b,d]|) proof let r be Point of I[01]; ::_thesis: for s being real number st r = s holds g2 . r = ((1 - ((2 * s) - 1)) * |[a,d]|) + (((2 * s) - 1) * |[b,d]|) let s be real number ; ::_thesis: ( r = s implies g2 . r = ((1 - ((2 * s) - 1)) * |[a,d]|) + (((2 * s) - 1) * |[b,d]|) ) assume A76: r = s ; ::_thesis: g2 . r = ((1 - ((2 * s) - 1)) * |[a,d]|) + (((2 * s) - 1) * |[b,d]|) g2 . r = ((h4 . r) * |[a,d]|) + ((h3 . r) * |[b,d]|) by A73 .= ((1 - ((h2 . r) - 1)) * |[a,d]|) + ((h3 . r) * |[b,d]|) by A65, A67 .= ((1 - ((h2 . r) - 1)) * |[a,d]|) + (((h2 . r) - 1) * |[b,d]|) by A65 .= ((1 - ((2 * (h1 . r)) - 1)) * |[a,d]|) + (((h2 . r) - 1) * |[b,d]|) by A61 .= ((1 - ((2 * (h1 . r)) - 1)) * |[a,d]|) + (((2 * (h1 . r)) - 1) * |[b,d]|) by A61 .= ((1 - ((2 * s) - 1)) * |[a,d]|) + (((2 * (h1 . r)) - 1) * |[b,d]|) by A59, A76 .= ((1 - ((2 * s) - 1)) * |[a,d]|) + (((2 * s) - 1) * |[b,d]|) by A59, A76 ; hence g2 . r = ((1 - ((2 * s) - 1)) * |[a,d]|) + (((2 * s) - 1) * |[b,d]|) ; ::_thesis: verum end; reconsider B11 = [.0,(1 / 2).] as non empty Subset of I[01] by A3, BORSUK_1:40, XBOOLE_1:7, XXREAL_1:1; A77: dom (g1 | B11) = (dom g1) /\ B11 by RELAT_1:61 .= the carrier of I[01] /\ B11 by FUNCT_2:def_1 .= B11 by XBOOLE_1:28 .= the carrier of (I[01] | B11) by PRE_TOPC:8 ; rng (g1 | B11) c= the carrier of (TOP-REAL 2) ; then reconsider g11 = g1 | B11 as Function of (I[01] | B11),(TOP-REAL 2) by A77, FUNCT_2:2; A78: TOP-REAL 2 is SubSpace of TOP-REAL 2 by TSEP_1:2; then A79: g11 is continuous by A70, BORSUK_4:44; reconsider B22 = [.(1 / 2),1.] as non empty Subset of I[01] by A3, BORSUK_1:40, XBOOLE_1:7, XXREAL_1:1; A80: dom (g2 | B22) = (dom g2) /\ B22 by RELAT_1:61 .= the carrier of I[01] /\ B22 by FUNCT_2:def_1 .= B22 by XBOOLE_1:28 .= the carrier of (I[01] | B22) by PRE_TOPC:8 ; rng (g2 | B22) c= the carrier of (TOP-REAL 2) ; then reconsider g22 = g2 | B22 as Function of (I[01] | B22),(TOP-REAL 2) by A80, FUNCT_2:2; A81: g22 is continuous by A74, A78, BORSUK_4:44; A82: B11 = [#] (I[01] | B11) by PRE_TOPC:def_5; A83: B22 = [#] (I[01] | B22) by PRE_TOPC:def_5; A84: B11 is closed by Th4; A85: B22 is closed by Th4; A86: ([#] (I[01] | B11)) \/ ([#] (I[01] | B22)) = [#] I[01] by A82, A83, BORSUK_1:40, XXREAL_1:165; for p being set st p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) holds g11 . p = g22 . p proof let p be set ; ::_thesis: ( p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) implies g11 . p = g22 . p ) assume A87: p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) ; ::_thesis: g11 . p = g22 . p then A88: p in [#] (I[01] | B11) by XBOOLE_0:def_4; A89: p in [#] (I[01] | B22) by A87; A90: p in B11 by A88, PRE_TOPC:def_5; A91: p in B22 by A89, PRE_TOPC:def_5; reconsider rp = p as Real by A90; A92: rp <= 1 / 2 by A90, XXREAL_1:1; rp >= 1 / 2 by A91, XXREAL_1:1; then rp = 1 / 2 by A92, XXREAL_0:1; then A93: 2 * rp = 1 ; thus g11 . p = g1 . p by A90, FUNCT_1:49 .= ((1 - 1) * |[a,c]|) + (1 * |[a,d]|) by A71, A90, A93 .= (0. (TOP-REAL 2)) + (1 * |[a,d]|) by EUCLID:29 .= ((1 - 0) * |[a,d]|) + ((1 - 1) * |[b,d]|) by EUCLID:29 .= g2 . p by A75, A90, A93 .= g22 . p by A91, FUNCT_1:49 ; ::_thesis: verum end; then consider h being Function of I[01],(TOP-REAL 2) such that A94: h = g11 +* g22 and A95: h is continuous by A79, A81, A82, A83, A84, A85, A86, JGRAPH_2:1; A96: dom f3 = dom h by Th5; A97: dom f3 = the carrier of I[01] by Th5; for x being set st x in dom f2 holds f3 . x = h . x proof let x be set ; ::_thesis: ( x in dom f2 implies f3 . x = h . x ) assume A98: x in dom f2 ; ::_thesis: f3 . x = h . x then reconsider rx = x as Real by A97, BORSUK_1:40; A99: 0 <= rx by A96, A98, BORSUK_1:40, XXREAL_1:1; A100: rx <= 1 by A96, A98, BORSUK_1:40, XXREAL_1:1; now__::_thesis:_(_(_rx_<_1_/_2_&_f3_._x_=_h_._x_)_or_(_rx_>=_1_/_2_&_f3_._x_=_h_._x_)_) percases ( rx < 1 / 2 or rx >= 1 / 2 ) ; caseA101: rx < 1 / 2 ; ::_thesis: f3 . x = h . x then A102: rx in [.0,(1 / 2).] by A99, XXREAL_1:1; not rx in dom g22 by A83, A101, XXREAL_1:1; then h . rx = g11 . rx by A94, FUNCT_4:11 .= g1 . rx by A102, FUNCT_1:49 .= ((1 - (2 * rx)) * |[a,c]|) + ((2 * rx) * |[a,d]|) by A71, A96, A98 .= f3 . rx by A32, A102 ; hence f3 . x = h . x ; ::_thesis: verum end; case rx >= 1 / 2 ; ::_thesis: f3 . x = h . x then A103: rx in [.(1 / 2),1.] by A100, XXREAL_1:1; then rx in [#] (I[01] | B22) by PRE_TOPC:def_5; then h . rx = g22 . rx by A80, A94, FUNCT_4:13 .= g2 . rx by A103, FUNCT_1:49 .= ((1 - ((2 * rx) - 1)) * |[a,d]|) + (((2 * rx) - 1) * |[b,d]|) by A75, A96, A98 .= f3 . rx by A35, A103 ; hence f3 . x = h . x ; ::_thesis: verum end; end; end; hence f3 . x = h . x ; ::_thesis: verum end; then A104: f2 = h by A96, FUNCT_1:2; A105: dom f3 = [#] I[01] by A14, BORSUK_1:40; for x1, x2 being set st x1 in dom f3 & x2 in dom f3 & f3 . x1 = f3 . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom f3 & x2 in dom f3 & f3 . x1 = f3 . x2 implies x1 = x2 ) assume that A106: x1 in dom f3 and A107: x2 in dom f3 and A108: f3 . x1 = f3 . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Real by A14, A106; reconsider r2 = x2 as Real by A14, A107; A109: (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) = {|[a,d]|} by A1, A2, Th34; now__::_thesis:_(_(_x1_in_[.0,(1_/_2).]_&_x2_in_[.0,(1_/_2).]_&_x1_=_x2_)_or_(_x1_in_[.0,(1_/_2).]_&_x2_in_[.(1_/_2),1.]_&_x1_=_x2_)_or_(_x1_in_[.(1_/_2),1.]_&_x2_in_[.0,(1_/_2).]_&_x1_=_x2_)_or_(_x1_in_[.(1_/_2),1.]_&_x2_in_[.(1_/_2),1.]_&_x1_=_x2_)_) percases ( ( x1 in [.0,(1 / 2).] & x2 in [.0,(1 / 2).] ) or ( x1 in [.0,(1 / 2).] & x2 in [.(1 / 2),1.] ) or ( x1 in [.(1 / 2),1.] & x2 in [.0,(1 / 2).] ) or ( x1 in [.(1 / 2),1.] & x2 in [.(1 / 2),1.] ) ) by A3, A14, A106, A107, XBOOLE_0:def_3; caseA110: ( x1 in [.0,(1 / 2).] & x2 in [.0,(1 / 2).] ) ; ::_thesis: x1 = x2 then f3 . r1 = ((1 - (2 * r1)) * |[a,c]|) + ((2 * r1) * |[a,d]|) by A32; then ((1 - (2 * r2)) * |[a,c]|) + ((2 * r2) * |[a,d]|) = ((1 - (2 * r1)) * |[a,c]|) + ((2 * r1) * |[a,d]|) by A32, A108, A110; then (((1 - (2 * r2)) * |[a,c]|) + ((2 * r2) * |[a,d]|)) - ((2 * r1) * |[a,d]|) = (1 - (2 * r1)) * |[a,c]| by EUCLID:48; then ((1 - (2 * r2)) * |[a,c]|) + (((2 * r2) * |[a,d]|) - ((2 * r1) * |[a,d]|)) = (1 - (2 * r1)) * |[a,c]| by EUCLID:45; then ((1 - (2 * r2)) * |[a,c]|) + (((2 * r2) - (2 * r1)) * |[a,d]|) = (1 - (2 * r1)) * |[a,c]| by EUCLID:50; then (((2 * r2) - (2 * r1)) * |[a,d]|) + (((1 - (2 * r2)) * |[a,c]|) - ((1 - (2 * r1)) * |[a,c]|)) = ((1 - (2 * r1)) * |[a,c]|) - ((1 - (2 * r1)) * |[a,c]|) by EUCLID:45; then (((2 * r2) - (2 * r1)) * |[a,d]|) + (((1 - (2 * r2)) * |[a,c]|) - ((1 - (2 * r1)) * |[a,c]|)) = 0. (TOP-REAL 2) by EUCLID:42; then (((2 * r2) - (2 * r1)) * |[a,d]|) + (((1 - (2 * r2)) - (1 - (2 * r1))) * |[a,c]|) = 0. (TOP-REAL 2) by EUCLID:50; then (((2 * r2) - (2 * r1)) * |[a,d]|) + ((- ((2 * r2) - (2 * r1))) * |[a,c]|) = 0. (TOP-REAL 2) ; then (((2 * r2) - (2 * r1)) * |[a,d]|) + (- (((2 * r2) - (2 * r1)) * |[a,c]|)) = 0. (TOP-REAL 2) by EUCLID:40; then (((2 * r2) - (2 * r1)) * |[a,d]|) - (((2 * r2) - (2 * r1)) * |[a,c]|) = 0. (TOP-REAL 2) by EUCLID:41; then ((2 * r2) - (2 * r1)) * (|[a,d]| - |[a,c]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( (2 * r2) - (2 * r1) = 0 or |[a,d]| - |[a,c]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( (2 * r2) - (2 * r1) = 0 or |[a,d]| = |[a,c]| ) by EUCLID:43; then ( (2 * r2) - (2 * r1) = 0 or d = |[a,c]| `2 ) by EUCLID:52; hence x1 = x2 by A2, EUCLID:52; ::_thesis: verum end; caseA111: ( x1 in [.0,(1 / 2).] & x2 in [.(1 / 2),1.] ) ; ::_thesis: x1 = x2 then A112: f3 . r1 = ((1 - (2 * r1)) * |[a,c]|) + ((2 * r1) * |[a,d]|) by A32; A113: 0 <= r1 by A111, XXREAL_1:1; r1 <= 1 / 2 by A111, XXREAL_1:1; then r1 * 2 <= (1 / 2) * 2 by XREAL_1:64; then A114: f3 . r1 in LSeg (|[a,c]|,|[a,d]|) by A112, A113; A115: f3 . r2 = ((1 - ((2 * r2) - 1)) * |[a,d]|) + (((2 * r2) - 1) * |[b,d]|) by A35, A111; A116: 1 / 2 <= r2 by A111, XXREAL_1:1; A117: r2 <= 1 by A111, XXREAL_1:1; r2 * 2 >= (1 / 2) * 2 by A116, XREAL_1:64; then A118: (2 * r2) - 1 >= 0 by XREAL_1:48; 2 * 1 >= 2 * r2 by A117, XREAL_1:64; then (1 + 1) - 1 >= (2 * r2) - 1 by XREAL_1:9; then f3 . r2 in { (((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by A115, A118; then f3 . r1 in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) by A108, A114, XBOOLE_0:def_4; then A119: f3 . r1 = |[a,d]| by A109, TARSKI:def_1; then (((1 - (2 * r1)) * |[a,c]|) + ((2 * r1) * |[a,d]|)) + (- |[a,d]|) = 0. (TOP-REAL 2) by A112, EUCLID:36; then (((1 - (2 * r1)) * |[a,c]|) + ((2 * r1) * |[a,d]|)) + ((- 1) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:39; then ((1 - (2 * r1)) * |[a,c]|) + (((2 * r1) * |[a,d]|) + ((- 1) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:26; then ((1 - (2 * r1)) * |[a,c]|) + (((2 * r1) + (- 1)) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:33; then ((1 - (2 * r1)) * |[a,c]|) + ((- (1 - (2 * r1))) * |[a,d]|) = 0. (TOP-REAL 2) ; then ((1 - (2 * r1)) * |[a,c]|) + (- ((1 - (2 * r1)) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:40; then ((1 - (2 * r1)) * |[a,c]|) - ((1 - (2 * r1)) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:41; then (1 - (2 * r1)) * (|[a,c]| - |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( 1 - (2 * r1) = 0 or |[a,c]| - |[a,d]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( 1 - (2 * r1) = 0 or |[a,c]| = |[a,d]| ) by EUCLID:43; then A120: ( 1 - (2 * r1) = 0 or c = |[a,d]| `2 ) by EUCLID:52; (((1 - ((2 * r2) - 1)) * |[a,d]|) + (((2 * r2) - 1) * |[b,d]|)) + (- |[a,d]|) = 0. (TOP-REAL 2) by A108, A115, A119, EUCLID:36; then (((1 - ((2 * r2) - 1)) * |[a,d]|) + (((2 * r2) - 1) * |[b,d]|)) + ((- 1) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:39; then (((2 * r2) - 1) * |[b,d]|) + (((1 - ((2 * r2) - 1)) * |[a,d]|) + ((- 1) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:26; then (((2 * r2) - 1) * |[b,d]|) + (((1 - ((2 * r2) - 1)) + (- 1)) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:33; then (((2 * r2) - 1) * |[b,d]|) + ((- ((2 * r2) - 1)) * |[a,d]|) = 0. (TOP-REAL 2) ; then (((2 * r2) - 1) * |[b,d]|) + (- (((2 * r2) - 1) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:40; then (((2 * r2) - 1) * |[b,d]|) - (((2 * r2) - 1) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:41; then ((2 * r2) - 1) * (|[b,d]| - |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( (2 * r2) - 1 = 0 or |[b,d]| - |[a,d]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( (2 * r2) - 1 = 0 or |[b,d]| = |[a,d]| ) by EUCLID:43; then ( (2 * r2) - 1 = 0 or b = |[a,d]| `1 ) by EUCLID:52; hence x1 = x2 by A1, A2, A120, EUCLID:52; ::_thesis: verum end; caseA121: ( x1 in [.(1 / 2),1.] & x2 in [.0,(1 / 2).] ) ; ::_thesis: x1 = x2 then A122: f3 . r2 = ((1 - (2 * r2)) * |[a,c]|) + ((2 * r2) * |[a,d]|) by A32; A123: 0 <= r2 by A121, XXREAL_1:1; r2 <= 1 / 2 by A121, XXREAL_1:1; then r2 * 2 <= (1 / 2) * 2 by XREAL_1:64; then A124: f3 . r2 in LSeg (|[a,c]|,|[a,d]|) by A122, A123; A125: f3 . r1 = ((1 - ((2 * r1) - 1)) * |[a,d]|) + (((2 * r1) - 1) * |[b,d]|) by A35, A121; A126: 1 / 2 <= r1 by A121, XXREAL_1:1; A127: r1 <= 1 by A121, XXREAL_1:1; r1 * 2 >= (1 / 2) * 2 by A126, XREAL_1:64; then A128: (2 * r1) - 1 >= 0 by XREAL_1:48; 2 * 1 >= 2 * r1 by A127, XREAL_1:64; then (1 + 1) - 1 >= (2 * r1) - 1 by XREAL_1:9; then f3 . r1 in { (((1 - lambda) * |[a,d]|) + (lambda * |[b,d]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by A125, A128; then f3 . r2 in (LSeg (|[a,c]|,|[a,d]|)) /\ (LSeg (|[a,d]|,|[b,d]|)) by A108, A124, XBOOLE_0:def_4; then A129: f3 . r2 = |[a,d]| by A109, TARSKI:def_1; then (((1 - (2 * r2)) * |[a,c]|) + ((2 * r2) * |[a,d]|)) + (- |[a,d]|) = 0. (TOP-REAL 2) by A122, EUCLID:36; then (((1 - (2 * r2)) * |[a,c]|) + ((2 * r2) * |[a,d]|)) + ((- 1) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:39; then ((1 - (2 * r2)) * |[a,c]|) + (((2 * r2) * |[a,d]|) + ((- 1) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:26; then ((1 - (2 * r2)) * |[a,c]|) + (((2 * r2) + (- 1)) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:33; then ((1 - (2 * r2)) * |[a,c]|) + ((- (1 - (2 * r2))) * |[a,d]|) = 0. (TOP-REAL 2) ; then ((1 - (2 * r2)) * |[a,c]|) + (- ((1 - (2 * r2)) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:40; then ((1 - (2 * r2)) * |[a,c]|) - ((1 - (2 * r2)) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:41; then (1 - (2 * r2)) * (|[a,c]| - |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( 1 - (2 * r2) = 0 or |[a,c]| - |[a,d]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( 1 - (2 * r2) = 0 or |[a,c]| = |[a,d]| ) by EUCLID:43; then A130: ( 1 - (2 * r2) = 0 or c = |[a,d]| `2 ) by EUCLID:52; (((1 - ((2 * r1) - 1)) * |[a,d]|) + (((2 * r1) - 1) * |[b,d]|)) + (- |[a,d]|) = 0. (TOP-REAL 2) by A108, A125, A129, EUCLID:36; then (((1 - ((2 * r1) - 1)) * |[a,d]|) + (((2 * r1) - 1) * |[b,d]|)) + ((- 1) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:39; then (((2 * r1) - 1) * |[b,d]|) + (((1 - ((2 * r1) - 1)) * |[a,d]|) + ((- 1) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:26; then (((2 * r1) - 1) * |[b,d]|) + (((- 1) + (1 - ((2 * r1) - 1))) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:33; then (((2 * r1) - 1) * |[b,d]|) + ((- ((2 * r1) - 1)) * |[a,d]|) = 0. (TOP-REAL 2) ; then (((2 * r1) - 1) * |[b,d]|) + (- (((2 * r1) - 1) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:40; then (((2 * r1) - 1) * |[b,d]|) - (((2 * r1) - 1) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:41; then ((2 * r1) - 1) * (|[b,d]| - |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( (2 * r1) - 1 = 0 or |[b,d]| - |[a,d]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( (2 * r1) - 1 = 0 or |[b,d]| = |[a,d]| ) by EUCLID:43; then ( (2 * r1) - 1 = 0 or b = |[a,d]| `1 ) by EUCLID:52; hence x1 = x2 by A1, A2, A130, EUCLID:52; ::_thesis: verum end; caseA131: ( x1 in [.(1 / 2),1.] & x2 in [.(1 / 2),1.] ) ; ::_thesis: x1 = x2 then f3 . r1 = ((1 - ((2 * r1) - 1)) * |[a,d]|) + (((2 * r1) - 1) * |[b,d]|) by A35; then ((1 - ((2 * r2) - 1)) * |[a,d]|) + (((2 * r2) - 1) * |[b,d]|) = ((1 - ((2 * r1) - 1)) * |[a,d]|) + (((2 * r1) - 1) * |[b,d]|) by A35, A108, A131; then (((1 - ((2 * r2) - 1)) * |[a,d]|) + (((2 * r2) - 1) * |[b,d]|)) - (((2 * r1) - 1) * |[b,d]|) = (1 - ((2 * r1) - 1)) * |[a,d]| by EUCLID:48; then ((1 - ((2 * r2) - 1)) * |[a,d]|) + ((((2 * r2) - 1) * |[b,d]|) - (((2 * r1) - 1) * |[b,d]|)) = (1 - ((2 * r1) - 1)) * |[a,d]| by EUCLID:45; then ((1 - ((2 * r2) - 1)) * |[a,d]|) + ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,d]|) = (1 - ((2 * r1) - 1)) * |[a,d]| by EUCLID:50; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,d]|) + (((1 - ((2 * r2) - 1)) * |[a,d]|) - ((1 - ((2 * r1) - 1)) * |[a,d]|)) = ((1 - ((2 * r1) - 1)) * |[a,d]|) - ((1 - ((2 * r1) - 1)) * |[a,d]|) by EUCLID:45; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,d]|) + (((1 - ((2 * r2) - 1)) * |[a,d]|) - ((1 - ((2 * r1) - 1)) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:42; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,d]|) + (((1 - ((2 * r2) - 1)) - (1 - ((2 * r1) - 1))) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:50; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,d]|) + ((- (((2 * r2) - 1) - ((2 * r1) - 1))) * |[a,d]|) = 0. (TOP-REAL 2) ; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,d]|) + (- ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,d]|)) = 0. (TOP-REAL 2) by EUCLID:40; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,d]|) - ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:41; then (((2 * r2) - 1) - ((2 * r1) - 1)) * (|[b,d]| - |[a,d]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or |[b,d]| - |[a,d]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or |[b,d]| = |[a,d]| ) by EUCLID:43; then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or b = |[a,d]| `1 ) by EUCLID:52; hence x1 = x2 by A1, EUCLID:52; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; then A132: f3 is one-to-one by FUNCT_1:def_4; [#] ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) c= rng f3 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in [#] ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) or y in rng f3 ) assume y in [#] ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) ; ::_thesis: y in rng f3 then A133: y in Upper_Arc (rectangle (a,b,c,d)) by PRE_TOPC:def_5; then reconsider q = y as Point of (TOP-REAL 2) ; A134: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; now__::_thesis:_(_(_q_in_LSeg_(|[a,c]|,|[a,d]|)_&_y_in_rng_f3_)_or_(_q_in_LSeg_(|[a,d]|,|[b,d]|)_&_y_in_rng_f3_)_) percases ( q in LSeg (|[a,c]|,|[a,d]|) or q in LSeg (|[a,d]|,|[b,d]|) ) by A133, A134, XBOOLE_0:def_3; caseA135: q in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: y in rng f3 then A136: 0 <= (((q `2) - c) / (d - c)) / 2 by A38; A137: (((q `2) - c) / (d - c)) / 2 <= 1 by A38, A135; A138: f3 . ((((q `2) - c) / (d - c)) / 2) = q by A38, A135; (((q `2) - c) / (d - c)) / 2 in [.0,1.] by A136, A137, XXREAL_1:1; hence y in rng f3 by A14, A138, FUNCT_1:def_3; ::_thesis: verum end; caseA139: q in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: y in rng f3 then A140: 0 <= ((((q `1) - a) / (b - a)) / 2) + (1 / 2) by A48; A141: ((((q `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 by A48, A139; A142: f3 . (((((q `1) - a) / (b - a)) / 2) + (1 / 2)) = q by A48, A139; ((((q `1) - a) / (b - a)) / 2) + (1 / 2) in [.0,1.] by A140, A141, XXREAL_1:1; hence y in rng f3 by A14, A142, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence y in rng f3 ; ::_thesis: verum end; then A143: rng f3 = [#] ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) by XBOOLE_0:def_10; I[01] is compact by HEINE:4, TOPMETR:20; then A144: f3 is being_homeomorphism by A95, A104, A105, A132, A143, COMPTS_1:17, JGRAPH_1:45; rng f3 = Upper_Arc (rectangle (a,b,c,d)) by A143, PRE_TOPC:def_5; hence ex f being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) st ( f is being_homeomorphism & f . 0 = W-min (rectangle (a,b,c,d)) & f . 1 = E-max (rectangle (a,b,c,d)) & rng f = Upper_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[a,c]|,|[a,d]|) holds ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f . ((((p `2) - c) / (d - c)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[a,d]|,|[b,d]|) holds ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) ) ) by A29, A31, A32, A35, A38, A48, A144; ::_thesis: verum end; theorem Th54: :: JGRAPH_6:54 for a, b, c, d being real number st a < b & c < d holds ex f being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) st ( f is being_homeomorphism & f . 0 = E-max (rectangle (a,b,c,d)) & f . 1 = W-min (rectangle (a,b,c,d)) & rng f = Lower_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[b,d]|,|[b,c]|) holds ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[b,c]|,|[a,c]|) holds ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) ) proof let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies ex f being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) st ( f is being_homeomorphism & f . 0 = E-max (rectangle (a,b,c,d)) & f . 1 = W-min (rectangle (a,b,c,d)) & rng f = Lower_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[b,d]|,|[b,c]|) holds ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[b,c]|,|[a,c]|) holds ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d ; ::_thesis: ex f being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) st ( f is being_homeomorphism & f . 0 = E-max (rectangle (a,b,c,d)) & f . 1 = W-min (rectangle (a,b,c,d)) & rng f = Lower_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[b,d]|,|[b,c]|) holds ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[b,c]|,|[a,c]|) holds ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) ) defpred S1[ set , set ] means for r being Real st $1 = r holds ( ( r in [.0,(1 / 2).] implies $2 = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( r in [.(1 / 2),1.] implies $2 = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) ); A3: [.0,1.] = [.0,(1 / 2).] \/ [.(1 / 2),1.] by XXREAL_1:165; A4: for x being set st x in [.0,1.] holds ex y being set st S1[x,y] proof let x be set ; ::_thesis: ( x in [.0,1.] implies ex y being set st S1[x,y] ) assume A5: x in [.0,1.] ; ::_thesis: ex y being set st S1[x,y] now__::_thesis:_(_(_x_in_[.0,(1_/_2).]_&_ex_y_being_set_st_S1[x,y]_)_or_(_x_in_[.(1_/_2),1.]_&_ex_y_being_set_st_S1[x,y]_)_) percases ( x in [.0,(1 / 2).] or x in [.(1 / 2),1.] ) by A3, A5, XBOOLE_0:def_3; caseA6: x in [.0,(1 / 2).] ; ::_thesis: ex y being set st S1[x,y] then reconsider r = x as Real ; A7: r <= 1 / 2 by A6, XXREAL_1:1; set y0 = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|); ( r in [.(1 / 2),1.] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) proof assume r in [.(1 / 2),1.] ; ::_thesis: ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) then 1 / 2 <= r by XXREAL_1:1; then A8: r = 1 / 2 by A7, XXREAL_0:1; then A9: ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = (0 * |[b,d]|) + |[b,c]| by EUCLID:29 .= (0. (TOP-REAL 2)) + |[b,c]| by EUCLID:29 .= |[b,c]| by EUCLID:27 ; ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = (1 * |[b,c]|) + (0. (TOP-REAL 2)) by A8, EUCLID:29 .= |[b,c]| + (0. (TOP-REAL 2)) by EUCLID:29 .= |[b,c]| by EUCLID:27 ; hence ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by A9; ::_thesis: verum end; then for r2 being Real st x = r2 holds ( ( r2 in [.0,(1 / 2).] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) ) & ( r2 in [.(1 / 2),1.] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) ) ) ; hence ex y being set st S1[x,y] ; ::_thesis: verum end; caseA10: x in [.(1 / 2),1.] ; ::_thesis: ex y being set st S1[x,y] then reconsider r = x as Real ; A11: 1 / 2 <= r by A10, XXREAL_1:1; set y0 = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|); ( r in [.0,(1 / 2).] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) proof assume r in [.0,(1 / 2).] ; ::_thesis: ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) then r <= 1 / 2 by XXREAL_1:1; then A12: r = 1 / 2 by A11, XXREAL_0:1; then A13: ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = |[b,c]| + (0 * |[a,c]|) by EUCLID:29 .= |[b,c]| + (0. (TOP-REAL 2)) by EUCLID:29 .= |[b,c]| by EUCLID:27 ; ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = (0. (TOP-REAL 2)) + (1 * |[b,c]|) by A12, EUCLID:29 .= (0. (TOP-REAL 2)) + |[b,c]| by EUCLID:29 .= |[b,c]| by EUCLID:27 ; hence ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by A13; ::_thesis: verum end; then for r2 being Real st x = r2 holds ( ( r2 in [.0,(1 / 2).] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) ) & ( r2 in [.(1 / 2),1.] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) ) ) ; hence ex y being set st S1[x,y] ; ::_thesis: verum end; end; end; hence ex y being set st S1[x,y] ; ::_thesis: verum end; ex f2 being Function st ( dom f2 = [.0,1.] & ( for x being set st x in [.0,1.] holds S1[x,f2 . x] ) ) from CLASSES1:sch_1(A4); then consider f2 being Function such that A14: dom f2 = [.0,1.] and A15: for x being set st x in [.0,1.] holds S1[x,f2 . x] ; rng f2 c= the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) ) assume y in rng f2 ; ::_thesis: y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) then consider x being set such that A16: x in dom f2 and A17: y = f2 . x by FUNCT_1:def_3; now__::_thesis:_(_(_x_in_[.0,(1_/_2).]_&_y_in_the_carrier_of_((TOP-REAL_2)_|_(Lower_Arc_(rectangle_(a,b,c,d))))_)_or_(_x_in_[.(1_/_2),1.]_&_y_in_the_carrier_of_((TOP-REAL_2)_|_(Lower_Arc_(rectangle_(a,b,c,d))))_)_) percases ( x in [.0,(1 / 2).] or x in [.(1 / 2),1.] ) by A3, A14, A16, XBOOLE_0:def_3; caseA18: x in [.0,(1 / 2).] ; ::_thesis: y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) then reconsider r = x as Real ; A19: 0 <= r by A18, XXREAL_1:1; r <= 1 / 2 by A18, XXREAL_1:1; then A20: r * 2 <= (1 / 2) * 2 by XREAL_1:64; f2 . x = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by A14, A15, A16, A18; then A21: y in LSeg (|[b,d]|,|[b,c]|) by A17, A19, A20; Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by A1, A2, Th52; then y in Lower_Arc (rectangle (a,b,c,d)) by A21, XBOOLE_0:def_3; hence y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) by PRE_TOPC:8; ::_thesis: verum end; caseA22: x in [.(1 / 2),1.] ; ::_thesis: y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) then reconsider r = x as Real ; A23: 1 / 2 <= r by A22, XXREAL_1:1; A24: r <= 1 by A22, XXREAL_1:1; r * 2 >= (1 / 2) * 2 by A23, XREAL_1:64; then A25: (2 * r) - 1 >= 0 by XREAL_1:48; 2 * 1 >= 2 * r by A24, XREAL_1:64; then A26: (1 + 1) - 1 >= (2 * r) - 1 by XREAL_1:9; f2 . x = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by A14, A15, A16, A22; then A27: y in LSeg (|[b,c]|,|[a,c]|) by A17, A25, A26; Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|)) by A1, A2, Th52; then y in Lower_Arc (rectangle (a,b,c,d)) by A27, XBOOLE_0:def_3; hence y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) by PRE_TOPC:8; ::_thesis: verum end; end; end; hence y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) ; ::_thesis: verum end; then reconsider f3 = f2 as Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) by A14, BORSUK_1:40, FUNCT_2:2; A28: 0 in [.0,1.] by XXREAL_1:1; 0 in [.0,(1 / 2).] by XXREAL_1:1; then A29: f3 . 0 = ((1 - (2 * 0)) * |[b,d]|) + ((2 * 0) * |[b,c]|) by A15, A28 .= (1 * |[b,d]|) + (0. (TOP-REAL 2)) by EUCLID:29 .= |[b,d]| + (0. (TOP-REAL 2)) by EUCLID:29 .= |[b,d]| by EUCLID:27 .= E-max (rectangle (a,b,c,d)) by A1, A2, Th46 ; A30: 1 in [.0,1.] by XXREAL_1:1; 1 in [.(1 / 2),1.] by XXREAL_1:1; then A31: f3 . 1 = ((1 - ((2 * 1) - 1)) * |[b,c]|) + (((2 * 1) - 1) * |[a,c]|) by A15, A30 .= (0 * |[b,c]|) + |[a,c]| by EUCLID:29 .= (0. (TOP-REAL 2)) + |[a,c]| by EUCLID:29 .= |[a,c]| by EUCLID:27 .= W-min (rectangle (a,b,c,d)) by A1, A2, Th46 ; A32: for r being Real st r in [.0,(1 / 2).] holds f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) proof let r be Real; ::_thesis: ( r in [.0,(1 / 2).] implies f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) assume A33: r in [.0,(1 / 2).] ; ::_thesis: f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) then A34: 0 <= r by XXREAL_1:1; r <= 1 / 2 by A33, XXREAL_1:1; then r <= 1 by XXREAL_0:2; then r in [.0,1.] by A34, XXREAL_1:1; hence f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by A15, A33; ::_thesis: verum end; A35: for r being Real st r in [.(1 / 2),1.] holds f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) proof let r be Real; ::_thesis: ( r in [.(1 / 2),1.] implies f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) assume A36: r in [.(1 / 2),1.] ; ::_thesis: f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) then A37: 1 / 2 <= r by XXREAL_1:1; r <= 1 by A36, XXREAL_1:1; then r in [.0,1.] by A37, XXREAL_1:1; hence f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by A15, A36; ::_thesis: verum end; A38: for p being Point of (TOP-REAL 2) st p in LSeg (|[b,d]|,|[b,c]|) holds ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2) - d) / (c - d)) / 2) = p ) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (|[b,d]|,|[b,c]|) implies ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2) - d) / (c - d)) / 2) = p ) ) assume A39: p in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2) - d) / (c - d)) / 2) = p ) A40: |[b,d]| `2 = d by EUCLID:52; A41: |[b,c]| `2 = c by EUCLID:52; then A42: c <= p `2 by A2, A39, A40, TOPREAL1:4; A43: p `2 <= d by A2, A39, A40, A41, TOPREAL1:4; d - c > 0 by A2, XREAL_1:50; then A44: - (d - c) < - 0 by XREAL_1:24; d - (p `2) >= 0 by A43, XREAL_1:48; then A45: - (d - (p `2)) <= - 0 ; (p `2) - d >= c - d by A42, XREAL_1:9; then ((p `2) - d) / (c - d) <= (c - d) / (c - d) by A44, XREAL_1:73; then ((p `2) - d) / (c - d) <= 1 by A44, XCMPLX_1:60; then A46: (((p `2) - d) / (c - d)) / 2 <= 1 / 2 by XREAL_1:72; set r = (((p `2) - d) / (c - d)) / 2; (((p `2) - d) / (c - d)) / 2 in [.0,(1 / 2).] by A44, A45, A46, XXREAL_1:1; then f3 . ((((p `2) - d) / (c - d)) / 2) = ((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * |[b,d]|) + ((2 * ((((p `2) - d) / (c - d)) / 2)) * |[b,c]|) by A32 .= |[((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * b),((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * d)]| + ((2 * ((((p `2) - d) / (c - d)) / 2)) * |[b,c]|) by EUCLID:58 .= |[((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * b),((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * d)]| + |[((2 * ((((p `2) - d) / (c - d)) / 2)) * b),((2 * ((((p `2) - d) / (c - d)) / 2)) * c)]| by EUCLID:58 .= |[(((1 * b) - ((2 * ((((p `2) - d) / (c - d)) / 2)) * b)) + ((2 * ((((p `2) - d) / (c - d)) / 2)) * b)),(((1 - (2 * ((((p `2) - d) / (c - d)) / 2))) * d) + ((2 * ((((p `2) - d) / (c - d)) / 2)) * c))]| by EUCLID:56 .= |[b,((1 * d) + ((((p `2) - d) / (c - d)) * (c - d)))]| .= |[b,((1 * d) + ((p `2) - d))]| by A44, XCMPLX_1:87 .= |[(p `1),(p `2)]| by A39, TOPREAL3:11 .= p by EUCLID:53 ; hence ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2) - d) / (c - d)) / 2) = p ) by A44, A45, A46, XXREAL_0:2; ::_thesis: verum end; A47: for p being Point of (TOP-REAL 2) st p in LSeg (|[b,c]|,|[a,c]|) holds ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (|[b,c]|,|[a,c]|) implies ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) assume A48: p in LSeg (|[b,c]|,|[a,c]|) ; ::_thesis: ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) A49: |[b,c]| `1 = b by EUCLID:52; A50: |[a,c]| `1 = a by EUCLID:52; then A51: a <= p `1 by A1, A48, A49, TOPREAL1:3; A52: p `1 <= b by A1, A48, A49, A50, TOPREAL1:3; b - a > 0 by A1, XREAL_1:50; then A53: - (b - a) < - 0 by XREAL_1:24; b - (p `1) >= 0 by A52, XREAL_1:48; then A54: - (b - (p `1)) <= - 0 ; then A55: ((((p `1) - b) / (a - b)) / 2) + (1 / 2) >= 0 + (1 / 2) by A53, XREAL_1:7; (p `1) - b >= a - b by A51, XREAL_1:9; then ((p `1) - b) / (a - b) <= (a - b) / (a - b) by A53, XREAL_1:73; then ((p `1) - b) / (a - b) <= 1 by A53, XCMPLX_1:60; then (((p `1) - b) / (a - b)) / 2 <= 1 / 2 by XREAL_1:72; then A56: ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= (1 / 2) + (1 / 2) by XREAL_1:7; set r = ((((p `1) - b) / (a - b)) / 2) + (1 / 2); ((((p `1) - b) / (a - b)) / 2) + (1 / 2) in [.(1 / 2),1.] by A55, A56, XXREAL_1:1; then f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = ((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * |[b,c]|) + (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * |[a,c]|) by A35 .= |[((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b),((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * c)]| + (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * |[a,c]|) by EUCLID:58 .= |[((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b),((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * c)]| + |[(((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * a),(((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c)]| by EUCLID:58 .= |[(((1 - ((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b) + (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * a)),(((1 * c) - (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c)) + (((2 * (((((p `1) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c))]| by EUCLID:56 .= |[((1 * b) + ((((p `1) - b) / (a - b)) * (a - b))),c]| .= |[((1 * b) + ((p `1) - b)),c]| by A53, XCMPLX_1:87 .= |[(p `1),(p `2)]| by A48, TOPREAL3:12 .= p by EUCLID:53 ; hence ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) by A53, A54, A56; ::_thesis: verum end; reconsider B00 = [.0,1.] as Subset of R^1 by TOPMETR:17; reconsider B01 = B00 as non empty Subset of R^1 by XXREAL_1:1; I[01] = R^1 | B01 by TOPMETR:19, TOPMETR:20; then consider h1 being Function of I[01],R^1 such that A57: for p being Point of I[01] holds h1 . p = p and A58: h1 is continuous by Th6; consider h2 being Function of I[01],R^1 such that A59: for p being Point of I[01] for r1 being real number st h1 . p = r1 holds h2 . p = 2 * r1 and A60: h2 is continuous by A58, JGRAPH_2:23; consider h5 being Function of I[01],R^1 such that A61: for p being Point of I[01] for r1 being real number st h2 . p = r1 holds h5 . p = 1 - r1 and A62: h5 is continuous by A60, Th8; consider h3 being Function of I[01],R^1 such that A63: for p being Point of I[01] for r1 being real number st h2 . p = r1 holds h3 . p = r1 - 1 and A64: h3 is continuous by A60, Th7; consider h4 being Function of I[01],R^1 such that A65: for p being Point of I[01] for r1 being real number st h3 . p = r1 holds h4 . p = 1 - r1 and A66: h4 is continuous by A64, Th8; consider g1 being Function of I[01],(TOP-REAL 2) such that A67: for r being Point of I[01] holds g1 . r = ((h5 . r) * |[b,d]|) + ((h2 . r) * |[b,c]|) and A68: g1 is continuous by A60, A62, Th13; A69: for r being Point of I[01] for s being real number st r = s holds g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) proof let r be Point of I[01]; ::_thesis: for s being real number st r = s holds g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) let s be real number ; ::_thesis: ( r = s implies g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) ) assume A70: r = s ; ::_thesis: g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) g1 . r = ((h5 . r) * |[b,d]|) + ((h2 . r) * |[b,c]|) by A67 .= ((1 - (2 * (h1 . r))) * |[b,d]|) + ((h2 . r) * |[b,c]|) by A59, A61 .= ((1 - (2 * (h1 . r))) * |[b,d]|) + ((2 * (h1 . r)) * |[b,c]|) by A59 .= ((1 - (2 * s)) * |[b,d]|) + ((2 * (h1 . r)) * |[b,c]|) by A57, A70 .= ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) by A57, A70 ; hence g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) ; ::_thesis: verum end; consider g2 being Function of I[01],(TOP-REAL 2) such that A71: for r being Point of I[01] holds g2 . r = ((h4 . r) * |[b,c]|) + ((h3 . r) * |[a,c]|) and A72: g2 is continuous by A64, A66, Th13; A73: for r being Point of I[01] for s being real number st r = s holds g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) proof let r be Point of I[01]; ::_thesis: for s being real number st r = s holds g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) let s be real number ; ::_thesis: ( r = s implies g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) ) assume A74: r = s ; ::_thesis: g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) g2 . r = ((h4 . r) * |[b,c]|) + ((h3 . r) * |[a,c]|) by A71 .= ((1 - ((h2 . r) - 1)) * |[b,c]|) + ((h3 . r) * |[a,c]|) by A63, A65 .= ((1 - ((h2 . r) - 1)) * |[b,c]|) + (((h2 . r) - 1) * |[a,c]|) by A63 .= ((1 - ((2 * (h1 . r)) - 1)) * |[b,c]|) + (((h2 . r) - 1) * |[a,c]|) by A59 .= ((1 - ((2 * (h1 . r)) - 1)) * |[b,c]|) + (((2 * (h1 . r)) - 1) * |[a,c]|) by A59 .= ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * (h1 . r)) - 1) * |[a,c]|) by A57, A74 .= ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) by A57, A74 ; hence g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) ; ::_thesis: verum end; reconsider B11 = [.0,(1 / 2).] as non empty Subset of I[01] by A3, BORSUK_1:40, XBOOLE_1:7, XXREAL_1:1; A75: dom (g1 | B11) = (dom g1) /\ B11 by RELAT_1:61 .= the carrier of I[01] /\ B11 by FUNCT_2:def_1 .= B11 by XBOOLE_1:28 .= the carrier of (I[01] | B11) by PRE_TOPC:8 ; rng (g1 | B11) c= the carrier of (TOP-REAL 2) ; then reconsider g11 = g1 | B11 as Function of (I[01] | B11),(TOP-REAL 2) by A75, FUNCT_2:2; A76: TOP-REAL 2 is SubSpace of TOP-REAL 2 by TSEP_1:2; then A77: g11 is continuous by A68, BORSUK_4:44; reconsider B22 = [.(1 / 2),1.] as non empty Subset of I[01] by A3, BORSUK_1:40, XBOOLE_1:7, XXREAL_1:1; A78: dom (g2 | B22) = (dom g2) /\ B22 by RELAT_1:61 .= the carrier of I[01] /\ B22 by FUNCT_2:def_1 .= B22 by XBOOLE_1:28 .= the carrier of (I[01] | B22) by PRE_TOPC:8 ; rng (g2 | B22) c= the carrier of (TOP-REAL 2) ; then reconsider g22 = g2 | B22 as Function of (I[01] | B22),(TOP-REAL 2) by A78, FUNCT_2:2; A79: g22 is continuous by A72, A76, BORSUK_4:44; A80: B11 = [#] (I[01] | B11) by PRE_TOPC:def_5; A81: B22 = [#] (I[01] | B22) by PRE_TOPC:def_5; A82: B11 is closed by Th4; A83: B22 is closed by Th4; A84: ([#] (I[01] | B11)) \/ ([#] (I[01] | B22)) = [#] I[01] by A80, A81, BORSUK_1:40, XXREAL_1:165; for p being set st p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) holds g11 . p = g22 . p proof let p be set ; ::_thesis: ( p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) implies g11 . p = g22 . p ) assume A85: p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) ; ::_thesis: g11 . p = g22 . p then A86: p in [#] (I[01] | B11) by XBOOLE_0:def_4; A87: p in [#] (I[01] | B22) by A85; A88: p in B11 by A86, PRE_TOPC:def_5; A89: p in B22 by A87, PRE_TOPC:def_5; reconsider rp = p as Real by A88; A90: rp <= 1 / 2 by A88, XXREAL_1:1; rp >= 1 / 2 by A89, XXREAL_1:1; then rp = 1 / 2 by A90, XXREAL_0:1; then A91: 2 * rp = 1 ; thus g11 . p = g1 . p by A88, FUNCT_1:49 .= ((1 - 1) * |[b,d]|) + (1 * |[b,c]|) by A69, A88, A91 .= (0. (TOP-REAL 2)) + (1 * |[b,c]|) by EUCLID:29 .= ((1 - 0) * |[b,c]|) + ((1 - 1) * |[a,c]|) by EUCLID:29 .= g2 . p by A73, A88, A91 .= g22 . p by A89, FUNCT_1:49 ; ::_thesis: verum end; then consider h being Function of I[01],(TOP-REAL 2) such that A92: h = g11 +* g22 and A93: h is continuous by A77, A79, A80, A81, A82, A83, A84, JGRAPH_2:1; A94: dom f3 = dom h by Th5; A95: dom f3 = the carrier of I[01] by Th5; for x being set st x in dom f2 holds f3 . x = h . x proof let x be set ; ::_thesis: ( x in dom f2 implies f3 . x = h . x ) assume A96: x in dom f2 ; ::_thesis: f3 . x = h . x then reconsider rx = x as Real by A95, BORSUK_1:40; A97: 0 <= rx by A94, A96, BORSUK_1:40, XXREAL_1:1; A98: rx <= 1 by A94, A96, BORSUK_1:40, XXREAL_1:1; percases ( rx < 1 / 2 or rx >= 1 / 2 ) ; supposeA99: rx < 1 / 2 ; ::_thesis: f3 . x = h . x then A100: rx in [.0,(1 / 2).] by A97, XXREAL_1:1; not rx in dom g22 by A81, A99, XXREAL_1:1; then h . rx = g11 . rx by A92, FUNCT_4:11 .= g1 . rx by A100, FUNCT_1:49 .= ((1 - (2 * rx)) * |[b,d]|) + ((2 * rx) * |[b,c]|) by A69, A94, A96 .= f3 . rx by A32, A100 ; hence f3 . x = h . x ; ::_thesis: verum end; suppose rx >= 1 / 2 ; ::_thesis: f3 . x = h . x then A101: rx in [.(1 / 2),1.] by A98, XXREAL_1:1; then rx in [#] (I[01] | B22) by PRE_TOPC:def_5; then h . rx = g22 . rx by A78, A92, FUNCT_4:13 .= g2 . rx by A101, FUNCT_1:49 .= ((1 - ((2 * rx) - 1)) * |[b,c]|) + (((2 * rx) - 1) * |[a,c]|) by A73, A94, A96 .= f3 . rx by A35, A101 ; hence f3 . x = h . x ; ::_thesis: verum end; end; end; then A102: f2 = h by A94, FUNCT_1:2; A103: dom f3 = [#] I[01] by A14, BORSUK_1:40; for x1, x2 being set st x1 in dom f3 & x2 in dom f3 & f3 . x1 = f3 . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom f3 & x2 in dom f3 & f3 . x1 = f3 . x2 implies x1 = x2 ) assume that A104: x1 in dom f3 and A105: x2 in dom f3 and A106: f3 . x1 = f3 . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Real by A14, A104; reconsider r2 = x2 as Real by A14, A105; A107: (LSeg (|[b,d]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[a,c]|)) = {|[b,c]|} by A1, A2, Th32; now__::_thesis:_(_(_x1_in_[.0,(1_/_2).]_&_x2_in_[.0,(1_/_2).]_&_x1_=_x2_)_or_(_x1_in_[.0,(1_/_2).]_&_x2_in_[.(1_/_2),1.]_&_x1_=_x2_)_or_(_x1_in_[.(1_/_2),1.]_&_x2_in_[.0,(1_/_2).]_&_x1_=_x2_)_or_(_x1_in_[.(1_/_2),1.]_&_x2_in_[.(1_/_2),1.]_&_x1_=_x2_)_) percases ( ( x1 in [.0,(1 / 2).] & x2 in [.0,(1 / 2).] ) or ( x1 in [.0,(1 / 2).] & x2 in [.(1 / 2),1.] ) or ( x1 in [.(1 / 2),1.] & x2 in [.0,(1 / 2).] ) or ( x1 in [.(1 / 2),1.] & x2 in [.(1 / 2),1.] ) ) by A3, A14, A104, A105, XBOOLE_0:def_3; caseA108: ( x1 in [.0,(1 / 2).] & x2 in [.0,(1 / 2).] ) ; ::_thesis: x1 = x2 then f3 . r1 = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by A32; then ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by A32, A106, A108; then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) - ((2 * r1) * |[b,c]|) = (1 - (2 * r1)) * |[b,d]| by EUCLID:48; then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) * |[b,c]|) - ((2 * r1) * |[b,c]|)) = (1 - (2 * r1)) * |[b,d]| by EUCLID:45; then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) - (2 * r1)) * |[b,c]|) = (1 - (2 * r1)) * |[b,d]| by EUCLID:50; then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|)) = ((1 - (2 * r1)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|) by EUCLID:45; then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|)) = 0. (TOP-REAL 2) by EUCLID:42; then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) - (1 - (2 * r1))) * |[b,d]|) = 0. (TOP-REAL 2) by EUCLID:50; then (((2 * r2) - (2 * r1)) * |[b,c]|) + ((- ((2 * r2) - (2 * r1))) * |[b,d]|) = 0. (TOP-REAL 2) ; then (((2 * r2) - (2 * r1)) * |[b,c]|) + (- (((2 * r2) - (2 * r1)) * |[b,d]|)) = 0. (TOP-REAL 2) by EUCLID:40; then (((2 * r2) - (2 * r1)) * |[b,c]|) - (((2 * r2) - (2 * r1)) * |[b,d]|) = 0. (TOP-REAL 2) by EUCLID:41; then ((2 * r2) - (2 * r1)) * (|[b,c]| - |[b,d]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( (2 * r2) - (2 * r1) = 0 or |[b,c]| - |[b,d]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( (2 * r2) - (2 * r1) = 0 or |[b,c]| = |[b,d]| ) by EUCLID:43; then ( (2 * r2) - (2 * r1) = 0 or d = |[b,c]| `2 ) by EUCLID:52; hence x1 = x2 by A2, EUCLID:52; ::_thesis: verum end; caseA109: ( x1 in [.0,(1 / 2).] & x2 in [.(1 / 2),1.] ) ; ::_thesis: x1 = x2 then A110: f3 . r1 = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by A32; A111: 0 <= r1 by A109, XXREAL_1:1; r1 <= 1 / 2 by A109, XXREAL_1:1; then r1 * 2 <= (1 / 2) * 2 by XREAL_1:64; then A112: f3 . r1 in LSeg (|[b,d]|,|[b,c]|) by A110, A111; A113: f3 . r2 = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) by A35, A109; A114: 1 / 2 <= r2 by A109, XXREAL_1:1; A115: r2 <= 1 by A109, XXREAL_1:1; r2 * 2 >= (1 / 2) * 2 by A114, XREAL_1:64; then A116: (2 * r2) - 1 >= 0 by XREAL_1:48; 2 * 1 >= 2 * r2 by A115, XREAL_1:64; then (1 + 1) - 1 >= (2 * r2) - 1 by XREAL_1:9; then f3 . r2 in { (((1 - lambda) * |[b,c]|) + (lambda * |[a,c]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by A113, A116; then f3 . r1 in (LSeg (|[b,d]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[a,c]|)) by A106, A112, XBOOLE_0:def_4; then A117: f3 . r1 = |[b,c]| by A107, TARSKI:def_1; then (((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|)) + (- |[b,c]|) = 0. (TOP-REAL 2) by A110, EUCLID:36; then (((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|)) + ((- 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:39; then ((1 - (2 * r1)) * |[b,d]|) + (((2 * r1) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:26; then ((1 - (2 * r1)) * |[b,d]|) + (((2 * r1) + (- 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:33; then ((1 - (2 * r1)) * |[b,d]|) + ((- (1 - (2 * r1))) * |[b,c]|) = 0. (TOP-REAL 2) ; then ((1 - (2 * r1)) * |[b,d]|) + (- ((1 - (2 * r1)) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:40; then ((1 - (2 * r1)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:41; then (1 - (2 * r1)) * (|[b,d]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( 1 - (2 * r1) = 0 or |[b,d]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( 1 - (2 * r1) = 0 or |[b,d]| = |[b,c]| ) by EUCLID:43; then A118: ( 1 - (2 * r1) = 0 or d = |[b,c]| `2 ) by EUCLID:52; (((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) + (- |[b,c]|) = 0. (TOP-REAL 2) by A106, A113, A117, EUCLID:36; then (((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) + ((- 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:39; then (((2 * r2) - 1) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:26; then (((2 * r2) - 1) * |[a,c]|) + (((1 - ((2 * r2) - 1)) + (- 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:33; then (((2 * r2) - 1) * |[a,c]|) + ((- ((2 * r2) - 1)) * |[b,c]|) = 0. (TOP-REAL 2) ; then (((2 * r2) - 1) * |[a,c]|) + (- (((2 * r2) - 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:40; then (((2 * r2) - 1) * |[a,c]|) - (((2 * r2) - 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:41; then ((2 * r2) - 1) * (|[a,c]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( (2 * r2) - 1 = 0 or |[a,c]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( (2 * r2) - 1 = 0 or |[a,c]| = |[b,c]| ) by EUCLID:43; then ( (2 * r2) - 1 = 0 or a = |[b,c]| `1 ) by EUCLID:52; hence x1 = x2 by A1, A2, A118, EUCLID:52; ::_thesis: verum end; caseA119: ( x1 in [.(1 / 2),1.] & x2 in [.0,(1 / 2).] ) ; ::_thesis: x1 = x2 then A120: f3 . r2 = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) by A32; A121: 0 <= r2 by A119, XXREAL_1:1; r2 <= 1 / 2 by A119, XXREAL_1:1; then r2 * 2 <= (1 / 2) * 2 by XREAL_1:64; then A122: f3 . r2 in LSeg (|[b,d]|,|[b,c]|) by A120, A121; A123: f3 . r1 = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by A35, A119; A124: 1 / 2 <= r1 by A119, XXREAL_1:1; A125: r1 <= 1 by A119, XXREAL_1:1; r1 * 2 >= (1 / 2) * 2 by A124, XREAL_1:64; then A126: (2 * r1) - 1 >= 0 by XREAL_1:48; 2 * 1 >= 2 * r1 by A125, XREAL_1:64; then (1 + 1) - 1 >= (2 * r1) - 1 by XREAL_1:9; then f3 . r1 in { (((1 - lambda) * |[b,c]|) + (lambda * |[a,c]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by A123, A126; then f3 . r2 in (LSeg (|[b,d]|,|[b,c]|)) /\ (LSeg (|[b,c]|,|[a,c]|)) by A106, A122, XBOOLE_0:def_4; then A127: f3 . r2 = |[b,c]| by A107, TARSKI:def_1; then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) + (- |[b,c]|) = 0. (TOP-REAL 2) by A120, EUCLID:36; then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) + ((- 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:39; then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:26; then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) + (- 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:33; then ((1 - (2 * r2)) * |[b,d]|) + ((- (1 - (2 * r2))) * |[b,c]|) = 0. (TOP-REAL 2) ; then ((1 - (2 * r2)) * |[b,d]|) + (- ((1 - (2 * r2)) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:40; then ((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r2)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:41; then (1 - (2 * r2)) * (|[b,d]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( 1 - (2 * r2) = 0 or |[b,d]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( 1 - (2 * r2) = 0 or |[b,d]| = |[b,c]| ) by EUCLID:43; then A128: ( 1 - (2 * r2) = 0 or d = |[b,c]| `2 ) by EUCLID:52; (((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|)) + (- |[b,c]|) = 0. (TOP-REAL 2) by A106, A123, A127, EUCLID:36; then (((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|)) + ((- 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:39; then (((2 * r1) - 1) * |[a,c]|) + (((1 - ((2 * r1) - 1)) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:26; then (((2 * r1) - 1) * |[a,c]|) + (((1 - ((2 * r1) - 1)) + (- 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:33; then (((2 * r1) - 1) * |[a,c]|) + ((- ((2 * r1) - 1)) * |[b,c]|) = 0. (TOP-REAL 2) ; then (((2 * r1) - 1) * |[a,c]|) + (- (((2 * r1) - 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:40; then (((2 * r1) - 1) * |[a,c]|) - (((2 * r1) - 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:41; then ((2 * r1) - 1) * (|[a,c]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( (2 * r1) - 1 = 0 or |[a,c]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( (2 * r1) - 1 = 0 or |[a,c]| = |[b,c]| ) by EUCLID:43; then ( (2 * r1) - 1 = 0 or a = |[b,c]| `1 ) by EUCLID:52; hence x1 = x2 by A1, A2, A128, EUCLID:52; ::_thesis: verum end; caseA129: ( x1 in [.(1 / 2),1.] & x2 in [.(1 / 2),1.] ) ; ::_thesis: x1 = x2 then f3 . r1 = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by A35; then ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by A35, A106, A129; then (((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) - (((2 * r1) - 1) * |[a,c]|) = (1 - ((2 * r1) - 1)) * |[b,c]| by EUCLID:48; then ((1 - ((2 * r2) - 1)) * |[b,c]|) + ((((2 * r2) - 1) * |[a,c]|) - (((2 * r1) - 1) * |[a,c]|)) = (1 - ((2 * r1) - 1)) * |[b,c]| by EUCLID:45; then ((1 - ((2 * r2) - 1)) * |[b,c]|) + ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) = (1 - ((2 * r1) - 1)) * |[b,c]| by EUCLID:50; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|)) = ((1 - ((2 * r1) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|) by EUCLID:45; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:42; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) - (1 - ((2 * r1) - 1))) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:50; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + ((- (((2 * r2) - 1) - ((2 * r1) - 1))) * |[b,c]|) = 0. (TOP-REAL 2) ; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (- ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:40; then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) - ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:41; then (((2 * r2) - 1) - ((2 * r1) - 1)) * (|[a,c]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:49; then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or |[a,c]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:31; then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or |[a,c]| = |[b,c]| ) by EUCLID:43; then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or a = |[b,c]| `1 ) by EUCLID:52; hence x1 = x2 by A1, EUCLID:52; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; then A130: f3 is one-to-one by FUNCT_1:def_4; [#] ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) c= rng f3 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in [#] ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) or y in rng f3 ) assume y in [#] ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) ; ::_thesis: y in rng f3 then A131: y in Lower_Arc (rectangle (a,b,c,d)) by PRE_TOPC:def_5; then reconsider q = y as Point of (TOP-REAL 2) ; A132: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; now__::_thesis:_(_(_q_in_LSeg_(|[b,d]|,|[b,c]|)_&_y_in_rng_f3_)_or_(_q_in_LSeg_(|[b,c]|,|[a,c]|)_&_y_in_rng_f3_)_) percases ( q in LSeg (|[b,d]|,|[b,c]|) or q in LSeg (|[b,c]|,|[a,c]|) ) by A131, A132, XBOOLE_0:def_3; caseA133: q in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: y in rng f3 then A134: 0 <= (((q `2) - d) / (c - d)) / 2 by A38; A135: (((q `2) - d) / (c - d)) / 2 <= 1 by A38, A133; A136: f3 . ((((q `2) - d) / (c - d)) / 2) = q by A38, A133; (((q `2) - d) / (c - d)) / 2 in [.0,1.] by A134, A135, XXREAL_1:1; hence y in rng f3 by A14, A136, FUNCT_1:def_3; ::_thesis: verum end; caseA137: q in LSeg (|[b,c]|,|[a,c]|) ; ::_thesis: y in rng f3 then A138: 0 <= ((((q `1) - b) / (a - b)) / 2) + (1 / 2) by A47; A139: ((((q `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 by A47, A137; A140: f3 . (((((q `1) - b) / (a - b)) / 2) + (1 / 2)) = q by A47, A137; ((((q `1) - b) / (a - b)) / 2) + (1 / 2) in [.0,1.] by A138, A139, XXREAL_1:1; hence y in rng f3 by A14, A140, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence y in rng f3 ; ::_thesis: verum end; then A141: rng f3 = [#] ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) by XBOOLE_0:def_10; I[01] is compact by HEINE:4, TOPMETR:20; then A142: f3 is being_homeomorphism by A93, A102, A103, A130, A141, COMPTS_1:17, JGRAPH_1:45; rng f3 = Lower_Arc (rectangle (a,b,c,d)) by A141, PRE_TOPC:def_5; hence ex f being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) st ( f is being_homeomorphism & f . 0 = E-max (rectangle (a,b,c,d)) & f . 1 = W-min (rectangle (a,b,c,d)) & rng f = Lower_Arc (rectangle (a,b,c,d)) & ( for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[b,d]|,|[b,c]|) holds ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg (|[b,c]|,|[a,c]|) holds ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) ) by A29, A31, A32, A35, A38, A47, A142; ::_thesis: verum end; theorem Th55: :: JGRAPH_6:55 for a, b, c, d being real number for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,c]|,|[a,d]|) & p2 in LSeg (|[a,c]|,|[a,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 <= p2 `2 ) proof let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,c]|,|[a,d]|) & p2 in LSeg (|[a,c]|,|[a,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 <= p2 `2 ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in LSeg (|[a,c]|,|[a,d]|) & p2 in LSeg (|[a,c]|,|[a,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 <= p2 `2 ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d and A3: p1 in LSeg (|[a,c]|,|[a,d]|) and A4: p2 in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 <= p2 `2 ) A5: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; A6: p1 `1 = a by A2, A3, Th1; A7: c <= p1 `2 by A2, A3, Th1; A8: p2 `1 = a by A2, A4, Th1; A9: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; A10: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; then A11: LSeg (|[a,c]|,|[a,d]|) c= Upper_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; A12: (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) = {(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d)))} by A5, JORDAN6:def_9; A13: now__::_thesis:_(_p2_in_Lower_Arc_(rectangle_(a,b,c,d))_implies_p2_=_W-min_(rectangle_(a,b,c,d))_) assume p2 in Lower_Arc (rectangle (a,b,c,d)) ; ::_thesis: p2 = W-min (rectangle (a,b,c,d)) then A14: p2 in (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) by A4, A11, XBOOLE_0:def_4; now__::_thesis:_not_p2_=_E-max_(rectangle_(a,b,c,d)) assume p2 = E-max (rectangle (a,b,c,d)) ; ::_thesis: contradiction then p2 `1 = b by A9, EUCLID:52; hence contradiction by A1, A4, TOPREAL3:11; ::_thesis: verum end; hence p2 = W-min (rectangle (a,b,c,d)) by A12, A14, TARSKI:def_2; ::_thesis: verum end; thus ( LE p1,p2, rectangle (a,b,c,d) implies p1 `2 <= p2 `2 ) ::_thesis: ( p1 `2 <= p2 `2 implies LE p1,p2, rectangle (a,b,c,d) ) proof assume LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: p1 `2 <= p2 `2 then A15: ( ( p1 in Upper_Arc (rectangle (a,b,c,d)) & p2 in Lower_Arc (rectangle (a,b,c,d)) & not p2 = W-min (rectangle (a,b,c,d)) ) or ( p1 in Upper_Arc (rectangle (a,b,c,d)) & p2 in Upper_Arc (rectangle (a,b,c,d)) & LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) ) or ( p1 in Lower_Arc (rectangle (a,b,c,d)) & p2 in Lower_Arc (rectangle (a,b,c,d)) & not p2 = W-min (rectangle (a,b,c,d)) & LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) ) ) by JORDAN6:def_10; consider f being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) such that A16: f is being_homeomorphism and A17: f . 0 = W-min (rectangle (a,b,c,d)) and A18: f . 1 = E-max (rectangle (a,b,c,d)) and rng f = Upper_Arc (rectangle (a,b,c,d)) and for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) and for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) and A19: for p being Point of (TOP-REAL 2) st p in LSeg (|[a,c]|,|[a,d]|) holds ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f . ((((p `2) - c) / (d - c)) / 2) = p ) and for p being Point of (TOP-REAL 2) st p in LSeg (|[a,d]|,|[b,d]|) holds ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) by A1, A2, Th53; reconsider s1 = (((p1 `2) - c) / (d - c)) / 2, s2 = (((p2 `2) - c) / (d - c)) / 2 as Real ; A20: f . s1 = p1 by A3, A19; A21: f . s2 = p2 by A4, A19; A22: d - c > 0 by A2, XREAL_1:50; A23: s1 <= 1 by A3, A19; A24: 0 <= s2 by A4, A19; s2 <= 1 by A4, A19; then s1 <= s2 by A13, A15, A16, A17, A18, A20, A21, A23, A24, JORDAN5C:def_3; then ((((p1 `2) - c) / (d - c)) / 2) * 2 <= ((((p2 `2) - c) / (d - c)) / 2) * 2 by XREAL_1:64; then (((p1 `2) - c) / (d - c)) * (d - c) <= (((p2 `2) - c) / (d - c)) * (d - c) by A22, XREAL_1:64; then (p1 `2) - c <= (((p2 `2) - c) / (d - c)) * (d - c) by A22, XCMPLX_1:87; then (p1 `2) - c <= (p2 `2) - c by A22, XCMPLX_1:87; then ((p1 `2) - c) + c <= ((p2 `2) - c) + c by XREAL_1:7; hence p1 `2 <= p2 `2 ; ::_thesis: verum end; thus ( p1 `2 <= p2 `2 implies LE p1,p2, rectangle (a,b,c,d) ) ::_thesis: verum proof assume A25: p1 `2 <= p2 `2 ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) for g being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A26: g is being_homeomorphism and A27: g . 0 = W-min (rectangle (a,b,c,d)) and g . 1 = E-max (rectangle (a,b,c,d)) and A28: g . s1 = p1 and A29: 0 <= s1 and A30: s1 <= 1 and A31: g . s2 = p2 and A32: 0 <= s2 and A33: s2 <= 1 ; ::_thesis: s1 <= s2 A34: dom g = the carrier of I[01] by FUNCT_2:def_1; A35: g is one-to-one by A26, TOPS_2:def_5; A36: the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) = Upper_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A26, TOPS_2:def_5; then A37: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A38: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A39: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A38, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A40: for p being Point of (TOP-REAL 2) for r1, r2 being real number st hh1 . p = r1 & hh2 . p = r2 holds h . p = r1 + r2 and A41: h is continuous by A39, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A42: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A43: s1 > s2 ; ::_thesis: contradiction A44: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A45: k . 0 = h . (W-min (rectangle (a,b,c,d))) by A27, A44, FUNCT_1:13 .= (h1 . (W-min (rectangle (a,b,c,d)))) + (h2 . (W-min (rectangle (a,b,c,d)))) by A40 .= ((W-min (rectangle (a,b,c,d))) `1) + (proj2 . (W-min (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((W-min (rectangle (a,b,c,d))) `1) + ((W-min (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= a + ((W-min (rectangle (a,b,c,d))) `2) by A42, EUCLID:52 .= a + c by A42, EUCLID:52 ; s1 in [.0,1.] by A29, A30, XXREAL_1:1; then A46: k . s1 = h . p1 by A28, A44, FUNCT_1:13 .= (h1 . p1) + (h2 . p1) by A40 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= a + (p1 `2) by A6, PSCOMP_1:def_6 ; A47: s2 in [.0,1.] by A32, A33, XXREAL_1:1; then A48: k . s2 = h . p2 by A31, A44, FUNCT_1:13 .= (h1 . p2) + (h2 . p2) by A40 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= a + (p2 `2) by A8, PSCOMP_1:def_6 ; A49: k . 0 <= k . s1 by A7, A45, A46, XREAL_1:7; A50: k . s1 <= k . s2 by A25, A46, A48, XREAL_1:7; A51: 0 in [.0,1.] by XXREAL_1:1; then A52: [.0,s2.] c= [.0,1.] by A47, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A47, A51, BORSUK_1:40, XXREAL_2:def_12; A53: B is connected by A32, A47, A51, BORSUK_1:40, BORSUK_4:24; A54: 0 in B by A32, XXREAL_1:1; A55: s2 in B by A32, XXREAL_1:1; A56: k . 0 is Real by XREAL_0:def_1; A57: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A58: xc in B and A59: k . xc = k . s1 by A37, A41, A49, A50, A53, A54, A55, A56, A57, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A60: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A61: x1 in dom k and A62: x2 in dom k and A63: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A61; reconsider r2 = x2 as Point of I[01] by A62; A64: k . x1 = h . (g1 . x1) by A61, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A40 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A65: k . x2 = h . (g1 . x2) by A62, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A40 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A66: g . r1 in Upper_Arc (rectangle (a,b,c,d)) by A36; A67: g . r2 in Upper_Arc (rectangle (a,b,c,d)) by A36; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A66; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A67; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[a,c]|,|[a,d]|)_&_g_._r2_in_LSeg_(|[a,c]|,|[a,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,c]|,|[a,d]|)_&_g_._r2_in_LSeg_(|[a,d]|,|[b,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,d]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[a,c]|,|[a,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,d]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[a,d]|,|[b,d]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ) by A10, A36, XBOOLE_0:def_3; caseA68: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; ::_thesis: x1 = x2 then A69: gr1 `1 = a by A2, Th1; gr2 `1 = a by A2, A68, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A63, A64, A65, A69, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A34, A35, FUNCT_1:def_4; ::_thesis: verum end; caseA70: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A71: gr1 `1 = a by A2, Th1; A72: gr1 `2 <= d by A2, A70, Th1; A73: gr2 `2 = d by A1, A70, Th3; A74: a <= gr2 `1 by A1, A70, Th3; A75: a + (gr1 `2) = (gr2 `1) + d by A1, A63, A64, A65, A70, A71, Th3; A76: now__::_thesis:_not_a_<>_gr2_`1 assume a <> gr2 `1 ; ::_thesis: contradiction then a < gr2 `1 by A74, XXREAL_0:1; hence contradiction by A72, A75, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_d assume gr1 `2 <> d ; ::_thesis: contradiction then d > gr1 `2 by A72, XXREAL_0:1; hence contradiction by A63, A64, A65, A71, A73, A74, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A71, A73, A76, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A34, A35, FUNCT_1:def_4; ::_thesis: verum end; caseA77: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; ::_thesis: x1 = x2 then A78: gr2 `1 = a by A2, Th1; A79: gr2 `2 <= d by A2, A77, Th1; A80: gr1 `2 = d by A1, A77, Th3; A81: a <= gr1 `1 by A1, A77, Th3; A82: a + (gr2 `2) = (gr1 `1) + d by A1, A63, A64, A65, A77, A78, Th3; A83: now__::_thesis:_not_a_<>_gr1_`1 assume a <> gr1 `1 ; ::_thesis: contradiction then a < gr1 `1 by A81, XXREAL_0:1; hence contradiction by A79, A82, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_d assume gr2 `2 <> d ; ::_thesis: contradiction then d > gr2 `2 by A79, XXREAL_0:1; hence contradiction by A63, A64, A65, A78, A80, A81, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A78, A80, A83, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A34, A35, FUNCT_1:def_4; ::_thesis: verum end; caseA84: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A85: gr1 `2 = d by A1, Th3; gr2 `2 = d by A1, A84, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A63, A64, A65, A85, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A34, A35, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A86: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A29, A30, XXREAL_1:1; then rxc = s1 by A52, A58, A59, A60, A86; hence contradiction by A43, A58, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) by A3, A4, A11, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A3, A4, A11, JORDAN6:def_10; ::_thesis: verum end; end; theorem Th56: :: JGRAPH_6:56 for a, b, c, d being real number for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) & p2 in LSeg (|[a,d]|,|[b,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff p1 `1 <= p2 `1 ) proof let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) & p2 in LSeg (|[a,d]|,|[b,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff p1 `1 <= p2 `1 ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) & p2 in LSeg (|[a,d]|,|[b,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff p1 `1 <= p2 `1 ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d and A3: p1 in LSeg (|[a,d]|,|[b,d]|) and A4: p2 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) iff p1 `1 <= p2 `1 ) A5: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; A6: p1 `2 = d by A1, A3, Th3; A7: a <= p1 `1 by A1, A3, Th3; A8: p1 `1 <= b by A1, A3, Th3; A9: p2 `2 = d by A1, A4, Th3; A10: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; A11: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; A12: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; then A13: LSeg (|[a,d]|,|[b,d]|) c= Upper_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; A14: (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) = {(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d)))} by A5, JORDAN6:def_9; A15: now__::_thesis:_(_p2_in_Lower_Arc_(rectangle_(a,b,c,d))_implies_p2_=_E-max_(rectangle_(a,b,c,d))_) assume p2 in Lower_Arc (rectangle (a,b,c,d)) ; ::_thesis: p2 = E-max (rectangle (a,b,c,d)) then A16: p2 in (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) by A4, A13, XBOOLE_0:def_4; now__::_thesis:_not_p2_=_W-min_(rectangle_(a,b,c,d)) assume p2 = W-min (rectangle (a,b,c,d)) ; ::_thesis: contradiction then p2 `2 = c by A10, EUCLID:52; hence contradiction by A2, A4, TOPREAL3:12; ::_thesis: verum end; hence p2 = E-max (rectangle (a,b,c,d)) by A14, A16, TARSKI:def_2; ::_thesis: verum end; thus ( LE p1,p2, rectangle (a,b,c,d) implies p1 `1 <= p2 `1 ) ::_thesis: ( p1 `1 <= p2 `1 implies LE p1,p2, rectangle (a,b,c,d) ) proof assume LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: p1 `1 <= p2 `1 then A17: ( ( p1 in Upper_Arc (rectangle (a,b,c,d)) & p2 in Lower_Arc (rectangle (a,b,c,d)) & not p2 = W-min (rectangle (a,b,c,d)) ) or ( p1 in Upper_Arc (rectangle (a,b,c,d)) & p2 in Upper_Arc (rectangle (a,b,c,d)) & LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) ) or ( p1 in Lower_Arc (rectangle (a,b,c,d)) & p2 in Lower_Arc (rectangle (a,b,c,d)) & not p2 = W-min (rectangle (a,b,c,d)) & LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) ) ) by JORDAN6:def_10; now__::_thesis:_(_(_p2_=_E-max_(rectangle_(a,b,c,d))_&_p1_`1_<=_p2_`1_)_or_(_p2_<>_E-max_(rectangle_(a,b,c,d))_&_p1_`1_<=_p2_`1_)_) percases ( p2 = E-max (rectangle (a,b,c,d)) or p2 <> E-max (rectangle (a,b,c,d)) ) ; case p2 = E-max (rectangle (a,b,c,d)) ; ::_thesis: p1 `1 <= p2 `1 hence p1 `1 <= p2 `1 by A8, A11, EUCLID:52; ::_thesis: verum end; caseA18: p2 <> E-max (rectangle (a,b,c,d)) ; ::_thesis: p1 `1 <= p2 `1 consider f being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) such that A19: f is being_homeomorphism and A20: f . 0 = W-min (rectangle (a,b,c,d)) and A21: f . 1 = E-max (rectangle (a,b,c,d)) and rng f = Upper_Arc (rectangle (a,b,c,d)) and for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[a,c]|) + ((2 * r) * |[a,d]|) and for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[a,d]|) + (((2 * r) - 1) * |[b,d]|) and for p being Point of (TOP-REAL 2) st p in LSeg (|[a,c]|,|[a,d]|) holds ( 0 <= (((p `2) - c) / (d - c)) / 2 & (((p `2) - c) / (d - c)) / 2 <= 1 & f . ((((p `2) - c) / (d - c)) / 2) = p ) and A22: for p being Point of (TOP-REAL 2) st p in LSeg (|[a,d]|,|[b,d]|) holds ( 0 <= ((((p `1) - a) / (b - a)) / 2) + (1 / 2) & ((((p `1) - a) / (b - a)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - a) / (b - a)) / 2) + (1 / 2)) = p ) by A1, A2, Th53; reconsider s1 = ((((p1 `1) - a) / (b - a)) / 2) + (1 / 2), s2 = ((((p2 `1) - a) / (b - a)) / 2) + (1 / 2) as Real ; A23: f . s1 = p1 by A3, A22; A24: f . s2 = p2 by A4, A22; A25: b - a > 0 by A1, XREAL_1:50; A26: s1 <= 1 by A3, A22; A27: 0 <= s2 by A4, A22; s2 <= 1 by A4, A22; then s1 <= s2 by A15, A17, A18, A19, A20, A21, A23, A24, A26, A27, JORDAN5C:def_3; then (((p1 `1) - a) / (b - a)) / 2 <= (((p2 `1) - a) / (b - a)) / 2 by XREAL_1:6; then ((((p1 `1) - a) / (b - a)) / 2) * 2 <= ((((p2 `1) - a) / (b - a)) / 2) * 2 by XREAL_1:64; then (((p1 `1) - a) / (b - a)) * (b - a) <= (((p2 `1) - a) / (b - a)) * (b - a) by A25, XREAL_1:64; then (p1 `1) - a <= (((p2 `1) - a) / (b - a)) * (b - a) by A25, XCMPLX_1:87; then (p1 `1) - a <= (p2 `1) - a by A25, XCMPLX_1:87; then ((p1 `1) - a) + a <= ((p2 `1) - a) + a by XREAL_1:7; hence p1 `1 <= p2 `1 ; ::_thesis: verum end; end; end; hence p1 `1 <= p2 `1 ; ::_thesis: verum end; thus ( p1 `1 <= p2 `1 implies LE p1,p2, rectangle (a,b,c,d) ) ::_thesis: verum proof assume A28: p1 `1 <= p2 `1 ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) for g being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A29: g is being_homeomorphism and A30: g . 0 = W-min (rectangle (a,b,c,d)) and g . 1 = E-max (rectangle (a,b,c,d)) and A31: g . s1 = p1 and A32: 0 <= s1 and A33: s1 <= 1 and A34: g . s2 = p2 and A35: 0 <= s2 and A36: s2 <= 1 ; ::_thesis: s1 <= s2 A37: dom g = the carrier of I[01] by FUNCT_2:def_1; A38: g is one-to-one by A29, TOPS_2:def_5; A39: the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) = Upper_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A29, TOPS_2:def_5; then A40: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A41: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A42: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A41, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A43: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A44: h is continuous by A42, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A45: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A46: s1 > s2 ; ::_thesis: contradiction A47: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A48: k . 0 = h . (W-min (rectangle (a,b,c,d))) by A30, A47, FUNCT_1:13 .= (h1 . (W-min (rectangle (a,b,c,d)))) + (h2 . (W-min (rectangle (a,b,c,d)))) by A43 .= ((W-min (rectangle (a,b,c,d))) `1) + (proj2 . (W-min (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((W-min (rectangle (a,b,c,d))) `1) + ((W-min (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= ((W-min (rectangle (a,b,c,d))) `1) + c by A45, EUCLID:52 .= a + c by A45, EUCLID:52 ; s1 in [.0,1.] by A32, A33, XXREAL_1:1; then A49: k . s1 = h . p1 by A31, A47, FUNCT_1:13 .= (h1 . p1) + (h2 . p1) by A43 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= (p1 `1) + d by A6, PSCOMP_1:def_6 ; A50: s2 in [.0,1.] by A35, A36, XXREAL_1:1; then A51: k . s2 = h . p2 by A34, A47, FUNCT_1:13 .= (h1 . p2) + (h2 . p2) by A43 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= (p2 `1) + d by A9, PSCOMP_1:def_6 ; A52: k . 0 <= k . s1 by A2, A7, A48, A49, XREAL_1:7; A53: k . s1 <= k . s2 by A28, A49, A51, XREAL_1:7; A54: 0 in [.0,1.] by XXREAL_1:1; then A55: [.0,s2.] c= [.0,1.] by A50, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A50, A54, BORSUK_1:40, XXREAL_2:def_12; A56: B is connected by A35, A50, A54, BORSUK_1:40, BORSUK_4:24; A57: 0 in B by A35, XXREAL_1:1; A58: s2 in B by A35, XXREAL_1:1; A59: k . 0 is Real by XREAL_0:def_1; A60: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A61: xc in B and A62: k . xc = k . s1 by A40, A44, A52, A53, A56, A57, A58, A59, A60, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A63: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A64: x1 in dom k and A65: x2 in dom k and A66: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A64; reconsider r2 = x2 as Point of I[01] by A65; A67: k . x1 = h . (g1 . x1) by A64, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A43 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A68: k . x2 = h . (g1 . x2) by A65, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A43 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A69: g . r1 in Upper_Arc (rectangle (a,b,c,d)) by A39; A70: g . r2 in Upper_Arc (rectangle (a,b,c,d)) by A39; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A69; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A70; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[a,c]|,|[a,d]|)_&_g_._r2_in_LSeg_(|[a,c]|,|[a,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,c]|,|[a,d]|)_&_g_._r2_in_LSeg_(|[a,d]|,|[b,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,d]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[a,c]|,|[a,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,d]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[a,d]|,|[b,d]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ) by A12, A39, XBOOLE_0:def_3; caseA71: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; ::_thesis: x1 = x2 then A72: gr1 `1 = a by A2, Th1; gr2 `1 = a by A2, A71, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A66, A67, A68, A72, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A37, A38, FUNCT_1:def_4; ::_thesis: verum end; caseA73: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A74: gr1 `1 = a by A2, Th1; A75: gr1 `2 <= d by A2, A73, Th1; A76: gr2 `2 = d by A1, A73, Th3; A77: a <= gr2 `1 by A1, A73, Th3; A78: a + (gr1 `2) = (gr2 `1) + d by A1, A66, A67, A68, A73, A74, Th3; A79: now__::_thesis:_not_a_<>_gr2_`1 assume a <> gr2 `1 ; ::_thesis: contradiction then a < gr2 `1 by A77, XXREAL_0:1; hence contradiction by A75, A78, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_d assume gr1 `2 <> d ; ::_thesis: contradiction then d > gr1 `2 by A75, XXREAL_0:1; hence contradiction by A66, A67, A68, A74, A76, A77, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A74, A76, A79, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A37, A38, FUNCT_1:def_4; ::_thesis: verum end; caseA80: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; ::_thesis: x1 = x2 then A81: gr2 `1 = a by A2, Th1; A82: gr2 `2 <= d by A2, A80, Th1; A83: gr1 `2 = d by A1, A80, Th3; A84: a <= gr1 `1 by A1, A80, Th3; A85: a + (gr2 `2) = (gr1 `1) + d by A1, A66, A67, A68, A80, A81, Th3; A86: now__::_thesis:_not_a_<>_gr1_`1 assume a <> gr1 `1 ; ::_thesis: contradiction then a < gr1 `1 by A84, XXREAL_0:1; hence contradiction by A82, A85, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_d assume gr2 `2 <> d ; ::_thesis: contradiction then d > gr2 `2 by A82, XXREAL_0:1; hence contradiction by A66, A67, A68, A81, A83, A84, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A81, A83, A86, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A37, A38, FUNCT_1:def_4; ::_thesis: verum end; caseA87: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A88: gr1 `2 = d by A1, Th3; gr2 `2 = d by A1, A87, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A66, A67, A68, A88, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A37, A38, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A89: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A32, A33, XXREAL_1:1; then rxc = s1 by A55, A61, A62, A63, A89; hence contradiction by A46, A61, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) by A3, A4, A13, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A3, A4, A13, JORDAN6:def_10; ::_thesis: verum end; end; theorem Th57: :: JGRAPH_6:57 for a, b, c, d being real number for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[b,c]|,|[b,d]|) & p2 in LSeg (|[b,c]|,|[b,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 >= p2 `2 ) proof let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[b,c]|,|[b,d]|) & p2 in LSeg (|[b,c]|,|[b,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 >= p2 `2 ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in LSeg (|[b,c]|,|[b,d]|) & p2 in LSeg (|[b,c]|,|[b,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 >= p2 `2 ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d and A3: p1 in LSeg (|[b,c]|,|[b,d]|) and A4: p2 in LSeg (|[b,c]|,|[b,d]|) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) iff p1 `2 >= p2 `2 ) A5: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; A6: p1 `1 = b by A2, A3, Th1; A7: p1 `2 <= d by A2, A3, Th1; A8: p2 `1 = b by A2, A4, Th1; A9: p2 `2 <= d by A2, A4, Th1; A10: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; A11: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; A12: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then A13: LSeg (|[b,d]|,|[b,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; A14: (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) = {(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d)))} by A5, JORDAN6:def_9; A15: now__::_thesis:_(_p1_in_Upper_Arc_(rectangle_(a,b,c,d))_implies_p1_=_E-max_(rectangle_(a,b,c,d))_) assume p1 in Upper_Arc (rectangle (a,b,c,d)) ; ::_thesis: p1 = E-max (rectangle (a,b,c,d)) then A16: p1 in (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) by A3, A13, XBOOLE_0:def_4; now__::_thesis:_not_p1_=_W-min_(rectangle_(a,b,c,d)) assume p1 = W-min (rectangle (a,b,c,d)) ; ::_thesis: contradiction then p1 `1 = a by A10, EUCLID:52; hence contradiction by A1, A3, TOPREAL3:11; ::_thesis: verum end; hence p1 = E-max (rectangle (a,b,c,d)) by A14, A16, TARSKI:def_2; ::_thesis: verum end; thus ( LE p1,p2, rectangle (a,b,c,d) implies p1 `2 >= p2 `2 ) ::_thesis: ( p1 `2 >= p2 `2 implies LE p1,p2, rectangle (a,b,c,d) ) proof assume LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: p1 `2 >= p2 `2 then A17: ( ( p1 in Upper_Arc (rectangle (a,b,c,d)) & p2 in Lower_Arc (rectangle (a,b,c,d)) & not p2 = W-min (rectangle (a,b,c,d)) ) or ( p1 in Upper_Arc (rectangle (a,b,c,d)) & p2 in Upper_Arc (rectangle (a,b,c,d)) & LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) ) or ( p1 in Lower_Arc (rectangle (a,b,c,d)) & p2 in Lower_Arc (rectangle (a,b,c,d)) & not p2 = W-min (rectangle (a,b,c,d)) & LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) ) ) by JORDAN6:def_10; now__::_thesis:_(_(_p1_=_E-max_(rectangle_(a,b,c,d))_&_p1_`2_>=_p2_`2_)_or_(_p1_<>_E-max_(rectangle_(a,b,c,d))_&_p1_`2_>=_p2_`2_)_) percases ( p1 = E-max (rectangle (a,b,c,d)) or p1 <> E-max (rectangle (a,b,c,d)) ) ; case p1 = E-max (rectangle (a,b,c,d)) ; ::_thesis: p1 `2 >= p2 `2 hence p1 `2 >= p2 `2 by A9, A11, EUCLID:52; ::_thesis: verum end; caseA18: p1 <> E-max (rectangle (a,b,c,d)) ; ::_thesis: p1 `2 >= p2 `2 consider f being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) such that A19: f is being_homeomorphism and A20: f . 0 = E-max (rectangle (a,b,c,d)) and A21: f . 1 = W-min (rectangle (a,b,c,d)) and rng f = Lower_Arc (rectangle (a,b,c,d)) and for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) and for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) and A22: for p being Point of (TOP-REAL 2) st p in LSeg (|[b,d]|,|[b,c]|) holds ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) and for p being Point of (TOP-REAL 2) st p in LSeg (|[b,c]|,|[a,c]|) holds ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) by A1, A2, Th54; reconsider s1 = (((p1 `2) - d) / (c - d)) / 2, s2 = (((p2 `2) - d) / (c - d)) / 2 as Real ; A23: f . s1 = p1 by A3, A22; A24: f . s2 = p2 by A4, A22; d - c > 0 by A2, XREAL_1:50; then A25: - (d - c) < - 0 by XREAL_1:24; A26: s1 <= 1 by A3, A22; A27: 0 <= s2 by A4, A22; s2 <= 1 by A4, A22; then s1 <= s2 by A15, A17, A18, A19, A20, A21, A23, A24, A26, A27, JORDAN5C:def_3; then ((((p1 `2) - d) / (c - d)) / 2) * 2 <= ((((p2 `2) - d) / (c - d)) / 2) * 2 by XREAL_1:64; then (((p1 `2) - d) / (c - d)) * (c - d) >= (((p2 `2) - d) / (c - d)) * (c - d) by A25, XREAL_1:65; then (p1 `2) - d >= (((p2 `2) - d) / (c - d)) * (c - d) by A25, XCMPLX_1:87; then (p1 `2) - d >= (p2 `2) - d by A25, XCMPLX_1:87; then ((p1 `2) - d) + d >= ((p2 `2) - d) + d by XREAL_1:7; hence p1 `2 >= p2 `2 ; ::_thesis: verum end; end; end; hence p1 `2 >= p2 `2 ; ::_thesis: verum end; thus ( p1 `2 >= p2 `2 implies LE p1,p2, rectangle (a,b,c,d) ) ::_thesis: verum proof assume A28: p1 `2 >= p2 `2 ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) now__::_thesis:_(_(_p2_=_W-min_(rectangle_(a,b,c,d))_&_contradiction_)_or_(_p2_<>_W-min_(rectangle_(a,b,c,d))_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_) percases ( p2 = W-min (rectangle (a,b,c,d)) or p2 <> W-min (rectangle (a,b,c,d)) ) ; case p2 = W-min (rectangle (a,b,c,d)) ; ::_thesis: contradiction then p2 = |[a,c]| by A1, A2, Th46; hence contradiction by A1, A8, EUCLID:52; ::_thesis: verum end; caseA29: p2 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A30: g is being_homeomorphism and A31: g . 0 = E-max (rectangle (a,b,c,d)) and g . 1 = W-min (rectangle (a,b,c,d)) and A32: g . s1 = p1 and A33: 0 <= s1 and A34: s1 <= 1 and A35: g . s2 = p2 and A36: 0 <= s2 and A37: s2 <= 1 ; ::_thesis: s1 <= s2 A38: dom g = the carrier of I[01] by FUNCT_2:def_1; A39: g is one-to-one by A30, TOPS_2:def_5; A40: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) = Lower_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A30, TOPS_2:def_5; then A41: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A42: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A43: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A42, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A44: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A45: h is continuous by A43, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A46: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A47: s1 > s2 ; ::_thesis: contradiction A48: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A49: k . 0 = h . (E-max (rectangle (a,b,c,d))) by A31, A48, FUNCT_1:13 .= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A44 .= ((E-max (rectangle (a,b,c,d))) `1) + (proj2 . (E-max (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((E-max (rectangle (a,b,c,d))) `1) + ((E-max (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= b + ((E-max (rectangle (a,b,c,d))) `2) by A46, EUCLID:52 .= b + d by A46, EUCLID:52 ; s1 in [.0,1.] by A33, A34, XXREAL_1:1; then A50: k . s1 = h . p1 by A32, A48, FUNCT_1:13 .= (h1 . p1) + (h2 . p1) by A44 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= b + (p1 `2) by A6, PSCOMP_1:def_6 ; A51: s2 in [.0,1.] by A36, A37, XXREAL_1:1; then A52: k . s2 = h . p2 by A35, A48, FUNCT_1:13 .= (proj1 . p2) + (proj2 . p2) by A44 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= b + (p2 `2) by A8, PSCOMP_1:def_6 ; A53: k . 0 >= k . s1 by A7, A49, A50, XREAL_1:7; A54: k . s1 >= k . s2 by A28, A50, A52, XREAL_1:7; A55: 0 in [.0,1.] by XXREAL_1:1; then A56: [.0,s2.] c= [.0,1.] by A51, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A51, A55, BORSUK_1:40, XXREAL_2:def_12; A57: B is connected by A36, A51, A55, BORSUK_1:40, BORSUK_4:24; A58: 0 in B by A36, XXREAL_1:1; A59: s2 in B by A36, XXREAL_1:1; A60: k . 0 is Real by XREAL_0:def_1; A61: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A62: xc in B and A63: k . xc = k . s1 by A41, A45, A53, A54, A57, A58, A59, A60, A61, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A64: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A65: x1 in dom k and A66: x2 in dom k and A67: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A65; reconsider r2 = x2 as Point of I[01] by A66; A68: k . x1 = h . (g1 . x1) by A65, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A44 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A69: k . x2 = h . (g1 . x2) by A66, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A44 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A70: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A40; A71: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A40; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A70; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A71; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[b,c]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[b,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[b,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[b,c]|,|[b,d]|) & g . r2 in LSeg (|[b,c]|,|[b,d]|) ) or ( g . r1 in LSeg (|[b,c]|,|[b,d]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[b,d]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ) by A12, A40, XBOOLE_0:def_3; caseA72: ( g . r1 in LSeg (|[b,c]|,|[b,d]|) & g . r2 in LSeg (|[b,c]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A73: gr1 `1 = b by A2, Th1; gr2 `1 = b by A2, A72, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A67, A68, A69, A73, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A38, A39, FUNCT_1:def_4; ::_thesis: verum end; caseA74: ( g . r1 in LSeg (|[b,c]|,|[b,d]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A75: gr1 `1 = b by A2, Th1; A76: c <= gr1 `2 by A2, A74, Th1; A77: gr2 `2 = c by A1, A74, Th3; A78: gr2 `1 <= b by A1, A74, Th3; A79: b + (gr1 `2) = (gr2 `1) + c by A1, A67, A68, A69, A74, A75, Th3; A80: now__::_thesis:_not_b_<>_gr2_`1 assume b <> gr2 `1 ; ::_thesis: contradiction then b > gr2 `1 by A78, XXREAL_0:1; hence contradiction by A76, A79, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_c assume gr1 `2 <> c ; ::_thesis: contradiction then c < gr1 `2 by A76, XXREAL_0:1; hence contradiction by A67, A68, A69, A75, A77, A78, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A75, A77, A80, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A38, A39, FUNCT_1:def_4; ::_thesis: verum end; caseA81: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A82: gr2 `1 = b by A2, Th1; A83: c <= gr2 `2 by A2, A81, Th1; A84: gr1 `2 = c by A1, A81, Th3; A85: gr1 `1 <= b by A1, A81, Th3; A86: b + (gr2 `2) = (gr1 `1) + c by A1, A67, A68, A69, A81, A82, Th3; A87: now__::_thesis:_not_b_<>_gr1_`1 assume b <> gr1 `1 ; ::_thesis: contradiction then b > gr1 `1 by A85, XXREAL_0:1; hence contradiction by A83, A86, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_c assume gr2 `2 <> c ; ::_thesis: contradiction then c < gr2 `2 by A83, XXREAL_0:1; hence contradiction by A67, A68, A69, A82, A84, A85, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A82, A84, A87, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A38, A39, FUNCT_1:def_4; ::_thesis: verum end; caseA88: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A89: gr1 `2 = c by A1, Th3; gr2 `2 = c by A1, A88, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A67, A68, A69, A89, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A38, A39, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A90: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A33, A34, XXREAL_1:1; then rxc = s1 by A56, A62, A63, A64, A90; hence contradiction by A47, A62, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A3, A4, A13, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A3, A4, A13, A29, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: verum end; end; theorem Th58: :: JGRAPH_6:58 for a, b, c, d being real number for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,c]|,|[b,c]|) & p2 in LSeg (|[a,c]|,|[b,c]|) holds ( LE p1,p2, rectangle (a,b,c,d) & p1 <> W-min (rectangle (a,b,c,d)) iff ( p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) proof let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,c]|,|[b,c]|) & p2 in LSeg (|[a,c]|,|[b,c]|) holds ( LE p1,p2, rectangle (a,b,c,d) & p1 <> W-min (rectangle (a,b,c,d)) iff ( p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in LSeg (|[a,c]|,|[b,c]|) & p2 in LSeg (|[a,c]|,|[b,c]|) implies ( LE p1,p2, rectangle (a,b,c,d) & p1 <> W-min (rectangle (a,b,c,d)) iff ( p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d and A3: p1 in LSeg (|[a,c]|,|[b,c]|) and A4: p2 in LSeg (|[a,c]|,|[b,c]|) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) & p1 <> W-min (rectangle (a,b,c,d)) iff ( p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) A5: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; A6: p1 `2 = c by A1, A3, Th3; A7: p1 `1 <= b by A1, A3, Th3; A8: p2 `2 = c by A1, A4, Th3; A9: a <= p2 `1 by A1, A4, Th3; A10: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; A11: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; A12: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then A13: LSeg (|[b,c]|,|[a,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; then A14: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3; A15: Lower_Arc (rectangle (a,b,c,d)) c= rectangle (a,b,c,d) by A5, JORDAN6:61; A16: (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) = {(W-min (rectangle (a,b,c,d))),(E-max (rectangle (a,b,c,d)))} by A5, JORDAN6:def_9; A17: now__::_thesis:_(_p1_in_Upper_Arc_(rectangle_(a,b,c,d))_implies_p1_=_W-min_(rectangle_(a,b,c,d))_) assume p1 in Upper_Arc (rectangle (a,b,c,d)) ; ::_thesis: p1 = W-min (rectangle (a,b,c,d)) then p1 in (Upper_Arc (rectangle (a,b,c,d))) /\ (Lower_Arc (rectangle (a,b,c,d))) by A3, A13, XBOOLE_0:def_4; then ( p1 = W-min (rectangle (a,b,c,d)) or p1 = E-max (rectangle (a,b,c,d)) ) by A16, TARSKI:def_2; hence p1 = W-min (rectangle (a,b,c,d)) by A2, A6, A11, EUCLID:52; ::_thesis: verum end; thus ( LE p1,p2, rectangle (a,b,c,d) & p1 <> W-min (rectangle (a,b,c,d)) implies ( p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) ::_thesis: ( p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) implies ( LE p1,p2, rectangle (a,b,c,d) & p1 <> W-min (rectangle (a,b,c,d)) ) ) proof assume that A18: LE p1,p2, rectangle (a,b,c,d) and A19: p1 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: ( p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) A20: ( ( p1 in Upper_Arc (rectangle (a,b,c,d)) & p2 in Lower_Arc (rectangle (a,b,c,d)) & not p2 = W-min (rectangle (a,b,c,d)) ) or ( p1 in Upper_Arc (rectangle (a,b,c,d)) & p2 in Upper_Arc (rectangle (a,b,c,d)) & LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) ) or ( p1 in Lower_Arc (rectangle (a,b,c,d)) & p2 in Lower_Arc (rectangle (a,b,c,d)) & not p2 = W-min (rectangle (a,b,c,d)) & LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) ) ) by A18, JORDAN6:def_10; consider f being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) such that A21: f is being_homeomorphism and A22: f . 0 = E-max (rectangle (a,b,c,d)) and A23: f . 1 = W-min (rectangle (a,b,c,d)) and rng f = Lower_Arc (rectangle (a,b,c,d)) and for r being Real st r in [.0,(1 / 2).] holds f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) and for r being Real st r in [.(1 / 2),1.] holds f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) and for p being Point of (TOP-REAL 2) st p in LSeg (|[b,d]|,|[b,c]|) holds ( 0 <= (((p `2) - d) / (c - d)) / 2 & (((p `2) - d) / (c - d)) / 2 <= 1 & f . ((((p `2) - d) / (c - d)) / 2) = p ) and A24: for p being Point of (TOP-REAL 2) st p in LSeg (|[b,c]|,|[a,c]|) holds ( 0 <= ((((p `1) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1) - b) / (a - b)) / 2) + (1 / 2)) = p ) by A1, A2, Th54; reconsider s1 = ((((p1 `1) - b) / (a - b)) / 2) + (1 / 2), s2 = ((((p2 `1) - b) / (a - b)) / 2) + (1 / 2) as Real ; A25: f . s1 = p1 by A3, A24; A26: f . s2 = p2 by A4, A24; b - a > 0 by A1, XREAL_1:50; then A27: - (b - a) < - 0 by XREAL_1:24; A28: s1 <= 1 by A3, A24; A29: 0 <= s2 by A4, A24; s2 <= 1 by A4, A24; then s1 <= s2 by A17, A19, A20, A21, A22, A23, A25, A26, A28, A29, JORDAN5C:def_3; then (((p1 `1) - b) / (a - b)) / 2 <= (((p2 `1) - b) / (a - b)) / 2 by XREAL_1:6; then ((((p1 `1) - b) / (a - b)) / 2) * 2 <= ((((p2 `1) - b) / (a - b)) / 2) * 2 by XREAL_1:64; then (((p1 `1) - b) / (a - b)) * (a - b) >= (((p2 `1) - b) / (a - b)) * (a - b) by A27, XREAL_1:65; then (p1 `1) - b >= (((p2 `1) - b) / (a - b)) * (a - b) by A27, XCMPLX_1:87; then (p1 `1) - b >= (p2 `1) - b by A27, XCMPLX_1:87; then ((p1 `1) - b) + b >= ((p2 `1) - b) + b by XREAL_1:7; hence p1 `1 >= p2 `1 ; ::_thesis: p2 <> W-min (rectangle (a,b,c,d)) now__::_thesis:_not_p2_=_W-min_(rectangle_(a,b,c,d)) assume A30: p2 = W-min (rectangle (a,b,c,d)) ; ::_thesis: contradiction then LE p2,p1, rectangle (a,b,c,d) by A5, A14, A15, JORDAN7:3; hence contradiction by A1, A2, A18, A19, A30, Th50, JORDAN6:57; ::_thesis: verum end; hence p2 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: verum end; thus ( p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) implies ( LE p1,p2, rectangle (a,b,c,d) & p1 <> W-min (rectangle (a,b,c,d)) ) ) ::_thesis: verum proof assume that A31: p1 `1 >= p2 `1 and A32: p2 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) & p1 <> W-min (rectangle (a,b,c,d)) ) A33: for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A34: g is being_homeomorphism and A35: g . 0 = E-max (rectangle (a,b,c,d)) and g . 1 = W-min (rectangle (a,b,c,d)) and A36: g . s1 = p1 and A37: 0 <= s1 and A38: s1 <= 1 and A39: g . s2 = p2 and A40: 0 <= s2 and A41: s2 <= 1 ; ::_thesis: s1 <= s2 A42: dom g = the carrier of I[01] by FUNCT_2:def_1; A43: g is one-to-one by A34, TOPS_2:def_5; A44: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) = Lower_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A34, TOPS_2:def_5; then A45: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A46: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A47: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A46, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A48: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A49: h is continuous by A47, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A50: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A51: s1 > s2 ; ::_thesis: contradiction A52: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A53: k . 0 = h . (E-max (rectangle (a,b,c,d))) by A35, A52, FUNCT_1:13 .= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A48 .= ((E-max (rectangle (a,b,c,d))) `1) + (proj2 . (E-max (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((E-max (rectangle (a,b,c,d))) `1) + ((E-max (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= ((E-max (rectangle (a,b,c,d))) `1) + d by A50, EUCLID:52 .= b + d by A50, EUCLID:52 ; s1 in [.0,1.] by A37, A38, XXREAL_1:1; then A54: k . s1 = h . p1 by A36, A52, FUNCT_1:13 .= (proj1 . p1) + (proj2 . p1) by A48 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= (p1 `1) + c by A6, PSCOMP_1:def_6 ; A55: s2 in [.0,1.] by A40, A41, XXREAL_1:1; then A56: k . s2 = h . p2 by A39, A52, FUNCT_1:13 .= (h1 . p2) + (h2 . p2) by A48 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= (p2 `1) + c by A8, PSCOMP_1:def_6 ; A57: k . 0 >= k . s1 by A2, A7, A53, A54, XREAL_1:7; A58: k . s1 >= k . s2 by A31, A54, A56, XREAL_1:7; A59: 0 in [.0,1.] by XXREAL_1:1; then A60: [.0,s2.] c= [.0,1.] by A55, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A55, A59, BORSUK_1:40, XXREAL_2:def_12; A61: B is connected by A40, A55, A59, BORSUK_1:40, BORSUK_4:24; A62: 0 in B by A40, XXREAL_1:1; A63: s2 in B by A40, XXREAL_1:1; A64: k . 0 is Real by XREAL_0:def_1; A65: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A66: xc in B and A67: k . xc = k . s1 by A45, A49, A57, A58, A61, A62, A63, A64, A65, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A68: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A69: x1 in dom k and A70: x2 in dom k and A71: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A69; reconsider r2 = x2 as Point of I[01] by A70; A72: k . x1 = h . (g1 . x1) by A69, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A48 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A73: k . x2 = h . (g1 . x2) by A70, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A48 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A74: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A44; A75: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A44; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A74; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A75; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ) by A12, A44, XBOOLE_0:def_3; caseA76: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A77: gr1 `1 = b by A2, Th1; gr2 `1 = b by A2, A76, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A71, A72, A73, A77, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A42, A43, FUNCT_1:def_4; ::_thesis: verum end; caseA78: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A79: gr1 `1 = b by A2, Th1; A80: c <= gr1 `2 by A2, A78, Th1; A81: gr2 `2 = c by A1, A78, Th3; A82: gr2 `1 <= b by A1, A78, Th3; A83: b + (gr1 `2) = (gr2 `1) + c by A1, A71, A72, A73, A78, A79, Th3; A84: now__::_thesis:_not_b_<>_gr2_`1 assume b <> gr2 `1 ; ::_thesis: contradiction then b > gr2 `1 by A82, XXREAL_0:1; hence contradiction by A80, A83, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_c assume gr1 `2 <> c ; ::_thesis: contradiction then c < gr1 `2 by A80, XXREAL_0:1; hence contradiction by A71, A72, A73, A79, A81, A82, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A79, A81, A84, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A42, A43, FUNCT_1:def_4; ::_thesis: verum end; caseA85: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A86: gr2 `1 = b by A2, Th1; A87: c <= gr2 `2 by A2, A85, Th1; A88: gr1 `2 = c by A1, A85, Th3; A89: gr1 `1 <= b by A1, A85, Th3; A90: b + (gr2 `2) = (gr1 `1) + c by A1, A71, A72, A73, A85, A86, Th3; A91: now__::_thesis:_not_b_<>_gr1_`1 assume b <> gr1 `1 ; ::_thesis: contradiction then b > gr1 `1 by A89, XXREAL_0:1; hence contradiction by A87, A90, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_c assume gr2 `2 <> c ; ::_thesis: contradiction then c < gr2 `2 by A87, XXREAL_0:1; hence contradiction by A71, A72, A73, A86, A88, A89, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A86, A88, A91, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A42, A43, FUNCT_1:def_4; ::_thesis: verum end; caseA92: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A93: gr1 `2 = c by A1, Th3; gr2 `2 = c by A1, A92, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A71, A72, A73, A93, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A42, A43, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A94: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A37, A38, XXREAL_1:1; then rxc = s1 by A60, A66, A67, A68, A94; hence contradiction by A51, A66, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; A95: now__::_thesis:_not_p1_=_W-min_(rectangle_(a,b,c,d)) assume A96: p1 = W-min (rectangle (a,b,c,d)) ; ::_thesis: contradiction then p1 `1 = a by A10, EUCLID:52; then p1 `1 = p2 `1 by A9, A31, XXREAL_0:1; then |[(p1 `1),(p1 `2)]| = p2 by A6, A8, EUCLID:53; hence contradiction by A32, A96, EUCLID:53; ::_thesis: verum end; LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A3, A4, A13, A33, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A3, A4, A13, A32, JORDAN6:def_10; ::_thesis: p1 <> W-min (rectangle (a,b,c,d)) thus p1 <> W-min (rectangle (a,b,c,d)) by A95; ::_thesis: verum end; end; theorem Th59: :: JGRAPH_6:59 for a, b, c, d being real number for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,c]|,|[a,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) proof let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,c]|,|[a,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in LSeg (|[a,c]|,|[a,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d and A3: p1 in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) A4: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; then A5: LSeg (|[a,c]|,|[a,d]|) c= Upper_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; A6: p1 `1 = a by A2, A3, Th1; A7: c <= p1 `2 by A2, A3, Th1; A8: p1 `2 <= d by A2, A3, Th1; thus ( not LE p1,p2, rectangle (a,b,c,d) or ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ::_thesis: ( ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) implies LE p1,p2, rectangle (a,b,c,d) ) proof assume A9: LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then A10: p1 in rectangle (a,b,c,d) by A4, JORDAN7:5; A11: p2 in rectangle (a,b,c,d) by A4, A9, JORDAN7:5; rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3 .= (((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|))) \/ (LSeg (|[b,c]|,|[a,c]|)) by XBOOLE_1:4 ; then ( p2 in ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|)) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A11, XBOOLE_0:def_3; then A12: ( p2 in (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by XBOOLE_0:def_3; now__::_thesis:_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_p1_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[a,d]|,|[b,d]|)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_p1_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[a,d]|,|[b,d]|)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_p1_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[a,d]|,|[b,d]|)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_p1_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[a,d]|,|[b,d]|)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_) percases ( p2 in LSeg (|[a,c]|,|[a,d]|) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A12, XBOOLE_0:def_3; case p2 in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A9, Th55; ::_thesis: verum end; case p2 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; case p2 in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; caseA13: p2 in LSeg (|[b,c]|,|[a,c]|) ; ::_thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) now__::_thesis:_(_(_p2_=_W-min_(rectangle_(a,b,c,d))_&_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_p1_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[a,d]|,|[b,d]|)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_<>_W-min_(rectangle_(a,b,c,d))_&_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_p1_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[a,d]|,|[b,d]|)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_) percases ( p2 = W-min (rectangle (a,b,c,d)) or p2 <> W-min (rectangle (a,b,c,d)) ) ; case p2 = W-min (rectangle (a,b,c,d)) ; ::_thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then LE p2,p1, rectangle (a,b,c,d) by A4, A10, JORDAN7:3; hence ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A9, Th50, JORDAN6:57; ::_thesis: verum end; case p2 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A13; ::_thesis: verum end; end; end; hence ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; end; end; hence ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; A14: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; thus ( ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) implies LE p1,p2, rectangle (a,b,c,d) ) ::_thesis: verum proof assume A15: ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) now__::_thesis:_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_p1_`2_<=_p2_`2_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_or_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_or_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_) percases ( ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A15; case ( p2 in LSeg (|[a,c]|,|[a,d]|) & p1 `2 <= p2 `2 ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) hence LE p1,p2, rectangle (a,b,c,d) by A1, A2, A3, Th55; ::_thesis: verum end; caseA16: p2 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) then A17: p2 `2 = d by A1, Th3; A18: a <= p2 `1 by A1, A16, Th3; A19: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; then A20: p2 in Upper_Arc (rectangle (a,b,c,d)) by A16, XBOOLE_0:def_3; A21: p1 in Upper_Arc (rectangle (a,b,c,d)) by A3, A19, XBOOLE_0:def_3; for g being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A22: g is being_homeomorphism and A23: g . 0 = W-min (rectangle (a,b,c,d)) and g . 1 = E-max (rectangle (a,b,c,d)) and A24: g . s1 = p1 and A25: 0 <= s1 and A26: s1 <= 1 and A27: g . s2 = p2 and A28: 0 <= s2 and A29: s2 <= 1 ; ::_thesis: s1 <= s2 A30: dom g = the carrier of I[01] by FUNCT_2:def_1; A31: g is one-to-one by A22, TOPS_2:def_5; A32: the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) = Upper_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A22, TOPS_2:def_5; then A33: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A34: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A35: h1 is continuous by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A34, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A36: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A37: h is continuous by A35, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A38: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A39: s1 > s2 ; ::_thesis: contradiction A40: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A41: k . 0 = h . (W-min (rectangle (a,b,c,d))) by A23, A40, FUNCT_1:13 .= (h1 . (W-min (rectangle (a,b,c,d)))) + (h2 . (W-min (rectangle (a,b,c,d)))) by A36 .= ((W-min (rectangle (a,b,c,d))) `1) + (proj2 . (W-min (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((W-min (rectangle (a,b,c,d))) `1) + ((W-min (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= ((W-min (rectangle (a,b,c,d))) `1) + c by A38, EUCLID:52 .= a + c by A38, EUCLID:52 ; s1 in [.0,1.] by A25, A26, XXREAL_1:1; then A42: k . s1 = h . p1 by A24, A40, FUNCT_1:13 .= (proj1 . p1) + (proj2 . p1) by A36 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= a + (p1 `2) by A6, PSCOMP_1:def_6 ; A43: s2 in [.0,1.] by A28, A29, XXREAL_1:1; then A44: k . s2 = h . p2 by A27, A40, FUNCT_1:13 .= (proj1 . p2) + (proj2 . p2) by A36 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= (p2 `1) + d by A17, PSCOMP_1:def_6 ; A45: k . 0 <= k . s1 by A7, A41, A42, XREAL_1:7; A46: k . s1 <= k . s2 by A8, A18, A42, A44, XREAL_1:7; A47: 0 in [.0,1.] by XXREAL_1:1; then A48: [.0,s2.] c= [.0,1.] by A43, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A43, A47, BORSUK_1:40, XXREAL_2:def_12; A49: B is connected by A28, A43, A47, BORSUK_1:40, BORSUK_4:24; A50: 0 in B by A28, XXREAL_1:1; A51: s2 in B by A28, XXREAL_1:1; A52: k . 0 is Real by XREAL_0:def_1; A53: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A54: xc in B and A55: k . xc = k . s1 by A33, A37, A45, A46, A49, A50, A51, A52, A53, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A56: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A57: x1 in dom k and A58: x2 in dom k and A59: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A57; reconsider r2 = x2 as Point of I[01] by A58; A60: k . x1 = h . (g1 . x1) by A57, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A36 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A61: k . x2 = h . (g1 . x2) by A58, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A36 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A62: g . r1 in Upper_Arc (rectangle (a,b,c,d)) by A32; A63: g . r2 in Upper_Arc (rectangle (a,b,c,d)) by A32; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A62; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A63; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[a,c]|,|[a,d]|)_&_g_._r2_in_LSeg_(|[a,c]|,|[a,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,c]|,|[a,d]|)_&_g_._r2_in_LSeg_(|[a,d]|,|[b,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,d]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[a,c]|,|[a,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,d]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[a,d]|,|[b,d]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ) by A19, A32, XBOOLE_0:def_3; caseA64: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; ::_thesis: x1 = x2 then A65: gr1 `1 = a by A2, Th1; gr2 `1 = a by A2, A64, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A59, A60, A61, A65, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A30, A31, FUNCT_1:def_4; ::_thesis: verum end; caseA66: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A67: gr1 `1 = a by A2, Th1; A68: gr1 `2 <= d by A2, A66, Th1; A69: gr2 `2 = d by A1, A66, Th3; A70: a <= gr2 `1 by A1, A66, Th3; A71: a + (gr1 `2) = (gr2 `1) + d by A1, A59, A60, A61, A66, A67, Th3; A72: now__::_thesis:_not_a_<>_gr2_`1 assume a <> gr2 `1 ; ::_thesis: contradiction then a < gr2 `1 by A70, XXREAL_0:1; hence contradiction by A68, A71, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_d assume gr1 `2 <> d ; ::_thesis: contradiction then d > gr1 `2 by A68, XXREAL_0:1; hence contradiction by A59, A60, A61, A67, A69, A70, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A67, A69, A72, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A30, A31, FUNCT_1:def_4; ::_thesis: verum end; caseA73: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; ::_thesis: x1 = x2 then A74: gr2 `1 = a by A2, Th1; A75: gr2 `2 <= d by A2, A73, Th1; A76: gr1 `2 = d by A1, A73, Th3; A77: a <= gr1 `1 by A1, A73, Th3; A78: a + (gr2 `2) = (gr1 `1) + d by A1, A59, A60, A61, A73, A74, Th3; A79: now__::_thesis:_not_a_<>_gr1_`1 assume a <> gr1 `1 ; ::_thesis: contradiction then a < gr1 `1 by A77, XXREAL_0:1; hence contradiction by A75, A78, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_d assume gr2 `2 <> d ; ::_thesis: contradiction then d > gr2 `2 by A75, XXREAL_0:1; hence contradiction by A59, A60, A61, A74, A76, A77, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A74, A76, A79, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A30, A31, FUNCT_1:def_4; ::_thesis: verum end; caseA80: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A81: gr1 `2 = d by A1, Th3; gr2 `2 = d by A1, A80, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A59, A60, A61, A81, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A30, A31, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A82: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A25, A26, XXREAL_1:1; then rxc = s1 by A48, A54, A55, A56, A82; hence contradiction by A39, A54, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) by A20, A21, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A20, A21, JORDAN6:def_10; ::_thesis: verum end; caseA83: p2 in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) then A84: p2 `1 = b by TOPREAL3:11; Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then A85: LSeg (|[b,d]|,|[b,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; p2 <> W-min (rectangle (a,b,c,d)) by A1, A14, A84, EUCLID:52; hence LE p1,p2, rectangle (a,b,c,d) by A3, A5, A83, A85, JORDAN6:def_10; ::_thesis: verum end; caseA86: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then LSeg (|[b,c]|,|[a,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; hence LE p1,p2, rectangle (a,b,c,d) by A3, A5, A86, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: verum end; end; theorem Th60: :: JGRAPH_6:60 for a, b, c, d being real number for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) proof let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in LSeg (|[a,d]|,|[b,d]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d and A3: p1 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) A4: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; then A5: LSeg (|[a,d]|,|[b,d]|) c= Upper_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; A6: p1 `2 = d by A1, A3, Th3; A7: a <= p1 `1 by A1, A3, Th3; thus ( not LE p1,p2, rectangle (a,b,c,d) or ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ::_thesis: ( ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) implies LE p1,p2, rectangle (a,b,c,d) ) proof assume A8: LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then A9: p1 in rectangle (a,b,c,d) by A4, JORDAN7:5; A10: p2 in rectangle (a,b,c,d) by A4, A8, JORDAN7:5; rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3 .= (((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|))) \/ (LSeg (|[b,c]|,|[a,c]|)) by XBOOLE_1:4 ; then ( p2 in ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|)) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A10, XBOOLE_0:def_3; then A11: ( p2 in (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by XBOOLE_0:def_3; now__::_thesis:_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_(_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_p1_`1_<=_p2_`1_)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_(_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_p1_`1_<=_p2_`1_)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_(_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_p1_`1_<=_p2_`1_)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_(_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_p1_`1_<=_p2_`1_)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_) percases ( p2 in LSeg (|[a,c]|,|[a,d]|) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A11, XBOOLE_0:def_3; case p2 in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, Th59; hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th50, JORDAN6:57; ::_thesis: verum end; case p2 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th56; ::_thesis: verum end; case p2 in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; caseA12: p2 in LSeg (|[b,c]|,|[a,c]|) ; ::_thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) now__::_thesis:_(_(_p2_=_W-min_(rectangle_(a,b,c,d))_&_(_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_p1_`1_<=_p2_`1_)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_<>_W-min_(rectangle_(a,b,c,d))_&_(_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_p1_`1_<=_p2_`1_)_or_p2_in_LSeg_(|[b,d]|,|[b,c]|)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_) percases ( p2 = W-min (rectangle (a,b,c,d)) or p2 <> W-min (rectangle (a,b,c,d)) ) ; case p2 = W-min (rectangle (a,b,c,d)) ; ::_thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then LE p2,p1, rectangle (a,b,c,d) by A4, A9, JORDAN7:3; hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th50, JORDAN6:57; ::_thesis: verum end; case p2 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A12; ::_thesis: verum end; end; end; hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; end; end; hence ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; A13: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; thus ( ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) implies LE p1,p2, rectangle (a,b,c,d) ) ::_thesis: verum proof assume A14: ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) now__::_thesis:_(_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_p1_`1_<=_p2_`1_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_or_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_) percases ( ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) or p2 in LSeg (|[b,d]|,|[b,c]|) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A14; caseA15: ( p2 in LSeg (|[a,d]|,|[b,d]|) & p1 `1 <= p2 `1 ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) then A16: p2 `2 = d by A1, Th3; A17: Upper_Arc (rectangle (a,b,c,d)) = (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) by A1, A2, Th51; then A18: p2 in Upper_Arc (rectangle (a,b,c,d)) by A15, XBOOLE_0:def_3; A19: p1 in Upper_Arc (rectangle (a,b,c,d)) by A3, A17, XBOOLE_0:def_3; for g being Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = W-min (rectangle (a,b,c,d)) & g . 1 = E-max (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A20: g is being_homeomorphism and A21: g . 0 = W-min (rectangle (a,b,c,d)) and g . 1 = E-max (rectangle (a,b,c,d)) and A22: g . s1 = p1 and A23: 0 <= s1 and A24: s1 <= 1 and A25: g . s2 = p2 and A26: 0 <= s2 and A27: s2 <= 1 ; ::_thesis: s1 <= s2 A28: dom g = the carrier of I[01] by FUNCT_2:def_1; A29: g is one-to-one by A20, TOPS_2:def_5; A30: the carrier of ((TOP-REAL 2) | (Upper_Arc (rectangle (a,b,c,d)))) = Upper_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A20, TOPS_2:def_5; then A31: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A32: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A33: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A32, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A34: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A35: h is continuous by A33, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A36: W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A37: s1 > s2 ; ::_thesis: contradiction A38: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A39: k . 0 = h . (W-min (rectangle (a,b,c,d))) by A21, A38, FUNCT_1:13 .= (h1 . (W-min (rectangle (a,b,c,d)))) + (h2 . (W-min (rectangle (a,b,c,d)))) by A34 .= ((W-min (rectangle (a,b,c,d))) `1) + (proj2 . (W-min (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((W-min (rectangle (a,b,c,d))) `1) + ((W-min (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= ((W-min (rectangle (a,b,c,d))) `1) + c by A36, EUCLID:52 .= a + c by A36, EUCLID:52 ; s1 in [.0,1.] by A23, A24, XXREAL_1:1; then A40: k . s1 = h . p1 by A22, A38, FUNCT_1:13 .= (proj1 . p1) + (proj2 . p1) by A34 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= (p1 `1) + d by A6, PSCOMP_1:def_6 ; A41: s2 in [.0,1.] by A26, A27, XXREAL_1:1; then A42: k . s2 = h . p2 by A25, A38, FUNCT_1:13 .= (proj1 . p2) + (proj2 . p2) by A34 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= (p2 `1) + d by A16, PSCOMP_1:def_6 ; A43: k . 0 <= k . s1 by A2, A7, A39, A40, XREAL_1:7; A44: k . s1 <= k . s2 by A15, A40, A42, XREAL_1:7; A45: 0 in [.0,1.] by XXREAL_1:1; then A46: [.0,s2.] c= [.0,1.] by A41, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A41, A45, BORSUK_1:40, XXREAL_2:def_12; A47: B is connected by A26, A41, A45, BORSUK_1:40, BORSUK_4:24; A48: 0 in B by A26, XXREAL_1:1; A49: s2 in B by A26, XXREAL_1:1; A50: k . 0 is Real by XREAL_0:def_1; A51: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A52: xc in B and A53: k . xc = k . s1 by A31, A35, A43, A44, A47, A48, A49, A50, A51, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A54: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A55: x1 in dom k and A56: x2 in dom k and A57: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A55; reconsider r2 = x2 as Point of I[01] by A56; A58: k . x1 = h . (g1 . x1) by A55, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A34 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A59: k . x2 = h . (g1 . x2) by A56, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A34 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A60: g . r1 in Upper_Arc (rectangle (a,b,c,d)) by A30; A61: g . r2 in Upper_Arc (rectangle (a,b,c,d)) by A30; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A60; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A61; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[a,c]|,|[a,d]|)_&_g_._r2_in_LSeg_(|[a,c]|,|[a,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,c]|,|[a,d]|)_&_g_._r2_in_LSeg_(|[a,d]|,|[b,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,d]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[a,c]|,|[a,d]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[a,d]|,|[b,d]|)_&_g_._r2_in_LSeg_(|[a,d]|,|[b,d]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) or ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ) by A17, A30, XBOOLE_0:def_3; caseA62: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; ::_thesis: x1 = x2 then A63: gr1 `1 = a by A2, Th1; gr2 `1 = a by A2, A62, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A57, A58, A59, A63, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A28, A29, FUNCT_1:def_4; ::_thesis: verum end; caseA64: ( g . r1 in LSeg (|[a,c]|,|[a,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A65: gr1 `1 = a by A2, Th1; A66: gr1 `2 <= d by A2, A64, Th1; A67: gr2 `2 = d by A1, A64, Th3; A68: a <= gr2 `1 by A1, A64, Th3; A69: a + (gr1 `2) = (gr2 `1) + d by A1, A57, A58, A59, A64, A65, Th3; A70: now__::_thesis:_not_a_<>_gr2_`1 assume a <> gr2 `1 ; ::_thesis: contradiction then a < gr2 `1 by A68, XXREAL_0:1; hence contradiction by A66, A69, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_d assume gr1 `2 <> d ; ::_thesis: contradiction then d > gr1 `2 by A66, XXREAL_0:1; hence contradiction by A57, A58, A59, A65, A67, A68, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A65, A67, A70, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A28, A29, FUNCT_1:def_4; ::_thesis: verum end; caseA71: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,c]|,|[a,d]|) ) ; ::_thesis: x1 = x2 then A72: gr2 `1 = a by A2, Th1; A73: gr2 `2 <= d by A2, A71, Th1; A74: gr1 `2 = d by A1, A71, Th3; A75: a <= gr1 `1 by A1, A71, Th3; A76: a + (gr2 `2) = (gr1 `1) + d by A1, A57, A58, A59, A71, A72, Th3; A77: now__::_thesis:_not_a_<>_gr1_`1 assume a <> gr1 `1 ; ::_thesis: contradiction then a < gr1 `1 by A75, XXREAL_0:1; hence contradiction by A73, A76, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_d assume gr2 `2 <> d ; ::_thesis: contradiction then d > gr2 `2 by A73, XXREAL_0:1; hence contradiction by A57, A58, A59, A72, A74, A75, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A72, A74, A77, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A28, A29, FUNCT_1:def_4; ::_thesis: verum end; caseA78: ( g . r1 in LSeg (|[a,d]|,|[b,d]|) & g . r2 in LSeg (|[a,d]|,|[b,d]|) ) ; ::_thesis: x1 = x2 then A79: gr1 `2 = d by A1, Th3; gr2 `2 = d by A1, A78, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A57, A58, A59, A79, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A28, A29, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A80: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A23, A24, XXREAL_1:1; then rxc = s1 by A46, A52, A53, A54, A80; hence contradiction by A37, A52, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Upper_Arc (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)) by A18, A19, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A18, A19, JORDAN6:def_10; ::_thesis: verum end; caseA81: p2 in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) then A82: p2 `1 = b by TOPREAL3:11; Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then A83: LSeg (|[b,d]|,|[b,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; p2 <> W-min (rectangle (a,b,c,d)) by A1, A13, A82, EUCLID:52; hence LE p1,p2, rectangle (a,b,c,d) by A3, A5, A81, A83, JORDAN6:def_10; ::_thesis: verum end; caseA84: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then LSeg (|[b,c]|,|[a,c]|) c= Lower_Arc (rectangle (a,b,c,d)) by XBOOLE_1:7; hence LE p1,p2, rectangle (a,b,c,d) by A3, A5, A84, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: verum end; end; theorem Th61: :: JGRAPH_6:61 for a, b, c, d being real number for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[b,d]|,|[b,c]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) proof let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[b,d]|,|[b,c]|) holds ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in LSeg (|[b,d]|,|[b,c]|) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d and A3: p1 in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) A4: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; A5: p1 `1 = b by A2, A3, Th1; A6: c <= p1 `2 by A2, A3, Th1; A7: p1 `2 <= d by A2, A3, Th1; thus ( not LE p1,p2, rectangle (a,b,c,d) or ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ::_thesis: ( ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) implies LE p1,p2, rectangle (a,b,c,d) ) proof assume A8: LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then A9: p1 in rectangle (a,b,c,d) by A4, JORDAN7:5; A10: p2 in rectangle (a,b,c,d) by A4, A8, JORDAN7:5; rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3 .= (((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|))) \/ (LSeg (|[b,c]|,|[a,c]|)) by XBOOLE_1:4 ; then ( p2 in ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|)) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A10, XBOOLE_0:def_3; then A11: ( p2 in (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by XBOOLE_0:def_3; now__::_thesis:_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_(_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_p1_`2_>=_p2_`2_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_(_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_p1_`2_>=_p2_`2_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_(_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_p1_`2_>=_p2_`2_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_(_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_p1_`2_>=_p2_`2_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_) percases ( p2 in LSeg (|[a,c]|,|[a,d]|) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A11, XBOOLE_0:def_3; case p2 in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, Th59; hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th50, JORDAN6:57; ::_thesis: verum end; case p2 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, Th60; hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th50, JORDAN6:57; ::_thesis: verum end; case p2 in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th57; ::_thesis: verum end; caseA12: p2 in LSeg (|[b,c]|,|[a,c]|) ; ::_thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) now__::_thesis:_(_(_p2_=_W-min_(rectangle_(a,b,c,d))_&_(_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_p1_`2_>=_p2_`2_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_or_(_p2_<>_W-min_(rectangle_(a,b,c,d))_&_(_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_p1_`2_>=_p2_`2_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_)_)_) percases ( p2 = W-min (rectangle (a,b,c,d)) or p2 <> W-min (rectangle (a,b,c,d)) ) ; case p2 = W-min (rectangle (a,b,c,d)) ; ::_thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) then LE p2,p1, rectangle (a,b,c,d) by A4, A9, JORDAN7:3; hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A1, A2, A3, A8, Th50, JORDAN6:57; ::_thesis: verum end; case p2 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A12; ::_thesis: verum end; end; end; hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; end; end; hence ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: verum end; thus ( ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) implies LE p1,p2, rectangle (a,b,c,d) ) ::_thesis: verum proof assume A13: ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) now__::_thesis:_(_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_p1_`2_>=_p2_`2_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_) percases ( ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A13; caseA14: ( p2 in LSeg (|[b,d]|,|[b,c]|) & p1 `2 >= p2 `2 ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) then A15: p2 `1 = b by A2, Th1; W-min (rectangle (a,b,c,d)) = |[a,c]| by A1, A2, Th46; then A16: p2 <> W-min (rectangle (a,b,c,d)) by A1, A15, EUCLID:52; A17: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then A18: p2 in Lower_Arc (rectangle (a,b,c,d)) by A14, XBOOLE_0:def_3; A19: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3, A17, XBOOLE_0:def_3; for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A20: g is being_homeomorphism and A21: g . 0 = E-max (rectangle (a,b,c,d)) and g . 1 = W-min (rectangle (a,b,c,d)) and A22: g . s1 = p1 and A23: 0 <= s1 and A24: s1 <= 1 and A25: g . s2 = p2 and A26: 0 <= s2 and A27: s2 <= 1 ; ::_thesis: s1 <= s2 A28: dom g = the carrier of I[01] by FUNCT_2:def_1; A29: g is one-to-one by A20, TOPS_2:def_5; A30: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) = Lower_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A20, TOPS_2:def_5; then A31: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A32: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A33: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A32, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A34: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A35: h is continuous by A33, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A36: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A37: s1 > s2 ; ::_thesis: contradiction A38: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A39: k . 0 = h . (E-max (rectangle (a,b,c,d))) by A21, A38, FUNCT_1:13 .= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A34 .= ((E-max (rectangle (a,b,c,d))) `1) + (proj2 . (E-max (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((E-max (rectangle (a,b,c,d))) `1) + ((E-max (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= ((E-max (rectangle (a,b,c,d))) `1) + d by A36, EUCLID:52 .= b + d by A36, EUCLID:52 ; s1 in [.0,1.] by A23, A24, XXREAL_1:1; then A40: k . s1 = h . p1 by A22, A38, FUNCT_1:13 .= (proj1 . p1) + (proj2 . p1) by A34 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= b + (p1 `2) by A5, PSCOMP_1:def_6 ; A41: s2 in [.0,1.] by A26, A27, XXREAL_1:1; then A42: k . s2 = h . p2 by A25, A38, FUNCT_1:13 .= (proj1 . p2) + (proj2 . p2) by A34 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= b + (p2 `2) by A15, PSCOMP_1:def_6 ; A43: k . 0 >= k . s1 by A7, A39, A40, XREAL_1:7; A44: k . s1 >= k . s2 by A14, A40, A42, XREAL_1:7; A45: 0 in [.0,1.] by XXREAL_1:1; then A46: [.0,s2.] c= [.0,1.] by A41, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A41, A45, BORSUK_1:40, XXREAL_2:def_12; A47: B is connected by A26, A41, A45, BORSUK_1:40, BORSUK_4:24; A48: 0 in B by A26, XXREAL_1:1; A49: s2 in B by A26, XXREAL_1:1; A50: k . 0 is Real by XREAL_0:def_1; A51: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A52: xc in B and A53: k . xc = k . s1 by A31, A35, A43, A44, A47, A48, A49, A50, A51, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A54: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A55: x1 in dom k and A56: x2 in dom k and A57: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A55; reconsider r2 = x2 as Point of I[01] by A56; A58: k . x1 = h . (g1 . x1) by A55, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A34 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A59: k . x2 = h . (g1 . x2) by A56, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A34 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A60: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A30; A61: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A30; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A60; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A61; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ) by A17, A30, XBOOLE_0:def_3; caseA62: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A63: gr1 `1 = b by A2, Th1; gr2 `1 = b by A2, A62, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A57, A58, A59, A63, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A28, A29, FUNCT_1:def_4; ::_thesis: verum end; caseA64: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A65: gr1 `1 = b by A2, Th1; A66: c <= gr1 `2 by A2, A64, Th1; A67: gr2 `2 = c by A1, A64, Th3; A68: gr2 `1 <= b by A1, A64, Th3; A69: b + (gr1 `2) = (gr2 `1) + c by A2, A57, A58, A59, A64, A67, Th1; A70: now__::_thesis:_not_b_<>_gr2_`1 assume b <> gr2 `1 ; ::_thesis: contradiction then b > gr2 `1 by A68, XXREAL_0:1; hence contradiction by A57, A58, A59, A65, A66, A67, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_c assume gr1 `2 <> c ; ::_thesis: contradiction then c < gr1 `2 by A66, XXREAL_0:1; hence contradiction by A68, A69, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A65, A67, A70, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A28, A29, FUNCT_1:def_4; ::_thesis: verum end; caseA71: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A72: gr2 `1 = b by A2, Th1; A73: c <= gr2 `2 by A2, A71, Th1; A74: gr1 `2 = c by A1, A71, Th3; A75: gr1 `1 <= b by A1, A71, Th3; A76: b + (gr2 `2) = (gr1 `1) + c by A1, A57, A58, A59, A71, A72, Th3; A77: now__::_thesis:_not_b_<>_gr1_`1 assume b <> gr1 `1 ; ::_thesis: contradiction then b > gr1 `1 by A75, XXREAL_0:1; hence contradiction by A73, A76, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_c assume gr2 `2 <> c ; ::_thesis: contradiction then c < gr2 `2 by A73, XXREAL_0:1; hence contradiction by A57, A58, A59, A72, A74, A75, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A72, A74, A77, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A28, A29, FUNCT_1:def_4; ::_thesis: verum end; caseA78: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A79: gr1 `2 = c by A1, Th3; gr2 `2 = c by A1, A78, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A57, A58, A59, A79, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A28, A29, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A80: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A23, A24, XXREAL_1:1; then rxc = s1 by A46, A52, A53, A54, A80; hence contradiction by A37, A52, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A18, A19, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A16, A18, A19, JORDAN6:def_10; ::_thesis: verum end; caseA81: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) then A82: p2 `2 = c by A1, Th3; A83: p2 `1 <= b by A1, A81, Th3; A84: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then A85: p2 in Lower_Arc (rectangle (a,b,c,d)) by A81, XBOOLE_0:def_3; A86: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3, A84, XBOOLE_0:def_3; for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A87: g is being_homeomorphism and A88: g . 0 = E-max (rectangle (a,b,c,d)) and g . 1 = W-min (rectangle (a,b,c,d)) and A89: g . s1 = p1 and A90: 0 <= s1 and A91: s1 <= 1 and A92: g . s2 = p2 and A93: 0 <= s2 and A94: s2 <= 1 ; ::_thesis: s1 <= s2 A95: dom g = the carrier of I[01] by FUNCT_2:def_1; A96: g is one-to-one by A87, TOPS_2:def_5; A97: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) = Lower_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A87, TOPS_2:def_5; then A98: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A99: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A100: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A99, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A101: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A102: h is continuous by A100, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A103: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A104: s1 > s2 ; ::_thesis: contradiction A105: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A106: k . 0 = h . (E-max (rectangle (a,b,c,d))) by A88, A105, FUNCT_1:13 .= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A101 .= ((E-max (rectangle (a,b,c,d))) `1) + (proj2 . (E-max (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((E-max (rectangle (a,b,c,d))) `1) + ((E-max (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= ((E-max (rectangle (a,b,c,d))) `1) + d by A103, EUCLID:52 .= b + d by A103, EUCLID:52 ; s1 in [.0,1.] by A90, A91, XXREAL_1:1; then A107: k . s1 = h . p1 by A89, A105, FUNCT_1:13 .= (proj1 . p1) + (proj2 . p1) by A101 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= b + (p1 `2) by A5, PSCOMP_1:def_6 ; A108: s2 in [.0,1.] by A93, A94, XXREAL_1:1; then A109: k . s2 = h . p2 by A92, A105, FUNCT_1:13 .= (proj1 . p2) + (proj2 . p2) by A101 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= (p2 `1) + c by A82, PSCOMP_1:def_6 ; A110: k . 0 >= k . s1 by A7, A106, A107, XREAL_1:7; A111: k . s1 >= k . s2 by A6, A83, A107, A109, XREAL_1:7; A112: 0 in [.0,1.] by XXREAL_1:1; then A113: [.0,s2.] c= [.0,1.] by A108, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A108, A112, BORSUK_1:40, XXREAL_2:def_12; A114: B is connected by A93, A108, A112, BORSUK_1:40, BORSUK_4:24; A115: 0 in B by A93, XXREAL_1:1; A116: s2 in B by A93, XXREAL_1:1; A117: k . 0 is Real by XREAL_0:def_1; A118: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A119: xc in B and A120: k . xc = k . s1 by A98, A102, A110, A111, A114, A115, A116, A117, A118, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A121: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A122: x1 in dom k and A123: x2 in dom k and A124: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A122; reconsider r2 = x2 as Point of I[01] by A123; A125: k . x1 = h . (g1 . x1) by A122, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A101 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A126: k . x2 = h . (g1 . x2) by A123, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A101 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A127: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A97; A128: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A97; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A127; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A128; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ) by A84, A97, XBOOLE_0:def_3; caseA129: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A130: gr1 `1 = b by A2, Th1; gr2 `1 = b by A2, A129, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A124, A125, A126, A130, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A95, A96, FUNCT_1:def_4; ::_thesis: verum end; caseA131: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A132: gr1 `1 = b by A2, Th1; A133: c <= gr1 `2 by A2, A131, Th1; A134: gr2 `2 = c by A1, A131, Th3; A135: gr2 `1 <= b by A1, A131, Th3; A136: b + (gr1 `2) = (gr2 `1) + c by A2, A124, A125, A126, A131, A134, Th1; A137: now__::_thesis:_not_b_<>_gr2_`1 assume b <> gr2 `1 ; ::_thesis: contradiction then b > gr2 `1 by A135, XXREAL_0:1; hence contradiction by A124, A125, A126, A132, A133, A134, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_c assume gr1 `2 <> c ; ::_thesis: contradiction then c < gr1 `2 by A133, XXREAL_0:1; hence contradiction by A135, A136, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A132, A134, A137, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A95, A96, FUNCT_1:def_4; ::_thesis: verum end; caseA138: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A139: gr2 `1 = b by A2, Th1; A140: c <= gr2 `2 by A2, A138, Th1; A141: gr1 `2 = c by A1, A138, Th3; A142: gr1 `1 <= b by A1, A138, Th3; A143: b + (gr2 `2) = (gr1 `1) + c by A1, A124, A125, A126, A138, A139, Th3; A144: now__::_thesis:_not_b_<>_gr1_`1 assume b <> gr1 `1 ; ::_thesis: contradiction then b > gr1 `1 by A142, XXREAL_0:1; hence contradiction by A140, A143, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_c assume gr2 `2 <> c ; ::_thesis: contradiction then c < gr2 `2 by A140, XXREAL_0:1; hence contradiction by A124, A125, A126, A139, A141, A142, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A139, A141, A144, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A95, A96, FUNCT_1:def_4; ::_thesis: verum end; caseA145: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A146: gr1 `2 = c by A1, Th3; gr2 `2 = c by A1, A145, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A124, A125, A126, A146, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A95, A96, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A147: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A90, A91, XXREAL_1:1; then rxc = s1 by A113, A119, A120, A121, A147; hence contradiction by A104, A119, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A85, A86, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A81, A85, A86, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: verum end; end; theorem Th62: :: JGRAPH_6:62 for a, b, c, d being real number for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[b,c]|,|[a,c]|) & p1 <> W-min (rectangle (a,b,c,d)) holds ( LE p1,p2, rectangle (a,b,c,d) iff ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) proof let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st a < b & c < d & p1 in LSeg (|[b,c]|,|[a,c]|) & p1 <> W-min (rectangle (a,b,c,d)) holds ( LE p1,p2, rectangle (a,b,c,d) iff ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in LSeg (|[b,c]|,|[a,c]|) & p1 <> W-min (rectangle (a,b,c,d)) implies ( LE p1,p2, rectangle (a,b,c,d) iff ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) ) set K = rectangle (a,b,c,d); assume that A1: a < b and A2: c < d and A3: p1 in LSeg (|[b,c]|,|[a,c]|) and A4: p1 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: ( LE p1,p2, rectangle (a,b,c,d) iff ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) A5: rectangle (a,b,c,d) is being_simple_closed_curve by A1, A2, Th50; A6: p1 `2 = c by A1, A3, Th3; A7: p1 `1 <= b by A1, A3, Th3; thus ( LE p1,p2, rectangle (a,b,c,d) implies ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ) ::_thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) implies LE p1,p2, rectangle (a,b,c,d) ) proof assume A8: LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) then A9: p2 in rectangle (a,b,c,d) by A5, JORDAN7:5; rectangle (a,b,c,d) = ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ ((LSeg (|[a,c]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[b,d]|))) by SPPOL_2:def_3 .= (((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|))) \/ (LSeg (|[b,c]|,|[a,c]|)) by XBOOLE_1:4 ; then ( p2 in ((LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|))) \/ (LSeg (|[b,d]|,|[b,c]|)) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A9, XBOOLE_0:def_3; then A10: ( p2 in (LSeg (|[a,c]|,|[a,d]|)) \/ (LSeg (|[a,d]|,|[b,d]|)) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by XBOOLE_0:def_3; now__::_thesis:_(_(_p2_in_LSeg_(|[a,c]|,|[a,d]|)_&_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p1_`1_>=_p2_`1_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_or_(_p2_in_LSeg_(|[a,d]|,|[b,d]|)_&_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p1_`1_>=_p2_`1_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_or_(_p2_in_LSeg_(|[b,d]|,|[b,c]|)_&_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p1_`1_>=_p2_`1_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p1_`1_>=_p2_`1_&_p2_<>_W-min_(rectangle_(a,b,c,d))_)_) percases ( p2 in LSeg (|[a,c]|,|[a,d]|) or p2 in LSeg (|[a,d]|,|[b,d]|) or p2 in LSeg (|[b,d]|,|[b,c]|) or p2 in LSeg (|[b,c]|,|[a,c]|) ) by A10, XBOOLE_0:def_3; case p2 in LSeg (|[a,c]|,|[a,d]|) ; ::_thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, A4, Th59; hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) by A1, A2, A3, A4, A8, Th50, JORDAN6:57; ::_thesis: verum end; case p2 in LSeg (|[a,d]|,|[b,d]|) ; ::_thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, A4, Th60; hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) by A1, A2, A3, A4, A8, Th50, JORDAN6:57; ::_thesis: verum end; case p2 in LSeg (|[b,d]|,|[b,c]|) ; ::_thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) then LE p2,p1, rectangle (a,b,c,d) by A1, A2, A3, A4, Th61; hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) by A1, A2, A3, A4, A8, Th50, JORDAN6:57; ::_thesis: verum end; case p2 in LSeg (|[b,c]|,|[a,c]|) ; ::_thesis: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) by A1, A2, A3, A4, A8, Th58; ::_thesis: verum end; end; end; hence ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) ) ; ::_thesis: verum end; thus ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 & p2 <> W-min (rectangle (a,b,c,d)) implies LE p1,p2, rectangle (a,b,c,d) ) ::_thesis: verum proof assume that A11: p2 in LSeg (|[b,c]|,|[a,c]|) and A12: p1 `1 >= p2 `1 and A13: p2 <> W-min (rectangle (a,b,c,d)) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) now__::_thesis:_(_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p1_`1_>=_p2_`1_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_or_(_p2_in_LSeg_(|[b,c]|,|[a,c]|)_&_p2_<>_W-min_(rectangle_(a,b,c,d))_&_LE_p1,p2,_rectangle_(a,b,c,d)_)_) percases ( ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 ) or ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ) by A11, A12; caseA14: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p1 `1 >= p2 `1 ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) then A15: p2 `2 = c by A1, Th3; A16: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then A17: p2 in Lower_Arc (rectangle (a,b,c,d)) by A14, XBOOLE_0:def_3; A18: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3, A16, XBOOLE_0:def_3; for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A19: g is being_homeomorphism and A20: g . 0 = E-max (rectangle (a,b,c,d)) and g . 1 = W-min (rectangle (a,b,c,d)) and A21: g . s1 = p1 and A22: 0 <= s1 and A23: s1 <= 1 and A24: g . s2 = p2 and A25: 0 <= s2 and A26: s2 <= 1 ; ::_thesis: s1 <= s2 A27: dom g = the carrier of I[01] by FUNCT_2:def_1; A28: g is one-to-one by A19, TOPS_2:def_5; A29: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) = Lower_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A19, TOPS_2:def_5; then A30: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A31: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A32: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A31, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A33: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A34: h is continuous by A32, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A35: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A36: s1 > s2 ; ::_thesis: contradiction A37: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A38: k . 0 = h . (E-max (rectangle (a,b,c,d))) by A20, A37, FUNCT_1:13 .= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A33 .= ((E-max (rectangle (a,b,c,d))) `1) + (proj2 . (E-max (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((E-max (rectangle (a,b,c,d))) `1) + ((E-max (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= ((E-max (rectangle (a,b,c,d))) `1) + d by A35, EUCLID:52 .= b + d by A35, EUCLID:52 ; s1 in [.0,1.] by A22, A23, XXREAL_1:1; then A39: k . s1 = h . p1 by A21, A37, FUNCT_1:13 .= (proj1 . p1) + (proj2 . p1) by A33 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= (p1 `1) + c by A6, PSCOMP_1:def_6 ; A40: s2 in [.0,1.] by A25, A26, XXREAL_1:1; then A41: k . s2 = h . p2 by A24, A37, FUNCT_1:13 .= (proj1 . p2) + (proj2 . p2) by A33 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= (p2 `1) + c by A15, PSCOMP_1:def_6 ; A42: k . 0 >= k . s1 by A2, A7, A38, A39, XREAL_1:7; A43: k . s1 >= k . s2 by A14, A39, A41, XREAL_1:7; A44: 0 in [.0,1.] by XXREAL_1:1; then A45: [.0,s2.] c= [.0,1.] by A40, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A40, A44, BORSUK_1:40, XXREAL_2:def_12; A46: B is connected by A25, A40, A44, BORSUK_1:40, BORSUK_4:24; A47: 0 in B by A25, XXREAL_1:1; A48: s2 in B by A25, XXREAL_1:1; A49: k . 0 is Real by XREAL_0:def_1; A50: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A51: xc in B and A52: k . xc = k . s1 by A30, A34, A42, A43, A46, A47, A48, A49, A50, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A53: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A54: x1 in dom k and A55: x2 in dom k and A56: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A54; reconsider r2 = x2 as Point of I[01] by A55; A57: k . x1 = h . (g1 . x1) by A54, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A33 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A58: k . x2 = h . (g1 . x2) by A55, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A33 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A59: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A29; A60: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A29; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A59; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A60; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ) by A16, A29, XBOOLE_0:def_3; caseA61: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A62: gr1 `1 = b by A2, Th1; gr2 `1 = b by A2, A61, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A56, A57, A58, A62, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A27, A28, FUNCT_1:def_4; ::_thesis: verum end; caseA63: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A64: gr1 `1 = b by A2, Th1; A65: c <= gr1 `2 by A2, A63, Th1; A66: gr2 `2 = c by A1, A63, Th3; A67: gr2 `1 <= b by A1, A63, Th3; A68: b + (gr1 `2) = (gr2 `1) + c by A2, A56, A57, A58, A63, A66, Th1; A69: now__::_thesis:_not_b_<>_gr2_`1 assume b <> gr2 `1 ; ::_thesis: contradiction then b > gr2 `1 by A67, XXREAL_0:1; hence contradiction by A56, A57, A58, A64, A65, A66, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_c assume gr1 `2 <> c ; ::_thesis: contradiction then c < gr1 `2 by A65, XXREAL_0:1; hence contradiction by A67, A68, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A64, A66, A69, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A27, A28, FUNCT_1:def_4; ::_thesis: verum end; caseA70: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A71: gr2 `1 = b by A2, Th1; A72: c <= gr2 `2 by A2, A70, Th1; A73: gr1 `2 = c by A1, A70, Th3; A74: gr1 `1 <= b by A1, A70, Th3; A75: b + (gr2 `2) = (gr1 `1) + c by A1, A56, A57, A58, A70, A71, Th3; A76: now__::_thesis:_not_b_<>_gr1_`1 assume b <> gr1 `1 ; ::_thesis: contradiction then b > gr1 `1 by A74, XXREAL_0:1; hence contradiction by A72, A75, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_c assume gr2 `2 <> c ; ::_thesis: contradiction then c < gr2 `2 by A72, XXREAL_0:1; hence contradiction by A56, A57, A58, A71, A73, A74, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A71, A73, A76, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A27, A28, FUNCT_1:def_4; ::_thesis: verum end; caseA77: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A78: gr1 `2 = c by A1, Th3; gr2 `2 = c by A1, A77, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A56, A57, A58, A78, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A27, A28, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A79: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A22, A23, XXREAL_1:1; then rxc = s1 by A45, A51, A52, A53, A79; hence contradiction by A36, A51, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A17, A18, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A13, A17, A18, JORDAN6:def_10; ::_thesis: verum end; caseA80: ( p2 in LSeg (|[b,c]|,|[a,c]|) & p2 <> W-min (rectangle (a,b,c,d)) ) ; ::_thesis: LE p1,p2, rectangle (a,b,c,d) then A81: p2 `2 = c by A1, Th3; A82: Lower_Arc (rectangle (a,b,c,d)) = (LSeg (|[b,d]|,|[b,c]|)) \/ (LSeg (|[b,c]|,|[a,c]|)) by A1, A2, Th52; then A83: p2 in Lower_Arc (rectangle (a,b,c,d)) by A80, XBOOLE_0:def_3; A84: p1 in Lower_Arc (rectangle (a,b,c,d)) by A3, A82, XBOOLE_0:def_3; for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 proof let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2 let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max (rectangle (a,b,c,d)) & g . 1 = W-min (rectangle (a,b,c,d)) & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 ) assume that A85: g is being_homeomorphism and A86: g . 0 = E-max (rectangle (a,b,c,d)) and g . 1 = W-min (rectangle (a,b,c,d)) and A87: g . s1 = p1 and A88: 0 <= s1 and A89: s1 <= 1 and A90: g . s2 = p2 and A91: 0 <= s2 and A92: s2 <= 1 ; ::_thesis: s1 <= s2 A93: dom g = the carrier of I[01] by FUNCT_2:def_1; A94: g is one-to-one by A85, TOPS_2:def_5; A95: the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle (a,b,c,d)))) = Lower_Arc (rectangle (a,b,c,d)) by PRE_TOPC:8; then reconsider g1 = g as Function of I[01],(TOP-REAL 2) by FUNCT_2:7; g is continuous by A85, TOPS_2:def_5; then A96: g1 is continuous by PRE_TOPC:26; reconsider h1 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17; reconsider hh1 = h1 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; reconsider hh2 = h2 as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ; A97: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) | ([#] TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #)) by TSEP_1:3 .= TopStruct(# the carrier of ((TOP-REAL 2) | ([#] (TOP-REAL 2))), the topology of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) #) by PRE_TOPC:36 .= (TOP-REAL 2) | ([#] (TOP-REAL 2)) ; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies hh1 is continuous ) by JGRAPH_2:29; then A98: ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh1 . p = proj1 . p ) implies h1 is continuous ) by PRE_TOPC:32; ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies hh2 is continuous ) by A97, JGRAPH_2:30; then ( ( for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) holds hh2 . p = proj2 . p ) implies h2 is continuous ) by PRE_TOPC:32; then consider h being Function of (TOP-REAL 2),R^1 such that A99: for p being Point of (TOP-REAL 2) for r1, r2 being real number st h1 . p = r1 & h2 . p = r2 holds h . p = r1 + r2 and A100: h is continuous by A98, JGRAPH_2:19; reconsider k = h * g1 as Function of I[01],R^1 ; A101: E-max (rectangle (a,b,c,d)) = |[b,d]| by A1, A2, Th46; now__::_thesis:_not_s1_>_s2 assume A102: s1 > s2 ; ::_thesis: contradiction A103: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then A104: k . 0 = h . (E-max (rectangle (a,b,c,d))) by A86, A103, FUNCT_1:13 .= (h1 . (E-max (rectangle (a,b,c,d)))) + (h2 . (E-max (rectangle (a,b,c,d)))) by A99 .= ((E-max (rectangle (a,b,c,d))) `1) + (proj2 . (E-max (rectangle (a,b,c,d)))) by PSCOMP_1:def_5 .= ((E-max (rectangle (a,b,c,d))) `1) + ((E-max (rectangle (a,b,c,d))) `2) by PSCOMP_1:def_6 .= ((E-max (rectangle (a,b,c,d))) `1) + d by A101, EUCLID:52 .= b + d by A101, EUCLID:52 ; s1 in [.0,1.] by A88, A89, XXREAL_1:1; then A105: k . s1 = h . p1 by A87, A103, FUNCT_1:13 .= (proj1 . p1) + (proj2 . p1) by A99 .= (p1 `1) + (proj2 . p1) by PSCOMP_1:def_5 .= (p1 `1) + c by A6, PSCOMP_1:def_6 ; A106: s2 in [.0,1.] by A91, A92, XXREAL_1:1; then A107: k . s2 = h . p2 by A90, A103, FUNCT_1:13 .= (proj1 . p2) + (proj2 . p2) by A99 .= (p2 `1) + (proj2 . p2) by PSCOMP_1:def_5 .= (p2 `1) + c by A81, PSCOMP_1:def_6 ; A108: k . 0 >= k . s1 by A2, A7, A104, A105, XREAL_1:7; A109: k . s1 >= k . s2 by A12, A105, A107, XREAL_1:7; A110: 0 in [.0,1.] by XXREAL_1:1; then A111: [.0,s2.] c= [.0,1.] by A106, XXREAL_2:def_12; reconsider B = [.0,s2.] as Subset of I[01] by A106, A110, BORSUK_1:40, XXREAL_2:def_12; A112: B is connected by A91, A106, A110, BORSUK_1:40, BORSUK_4:24; A113: 0 in B by A91, XXREAL_1:1; A114: s2 in B by A91, XXREAL_1:1; A115: k . 0 is Real by XREAL_0:def_1; A116: k . s2 is Real by XREAL_0:def_1; k . s1 is Real by XREAL_0:def_1; then consider xc being Point of I[01] such that A117: xc in B and A118: k . xc = k . s1 by A96, A100, A108, A109, A112, A113, A114, A115, A116, TOPREAL5:5; xc in [.0,1.] by BORSUK_1:40; then reconsider rxc = xc as Real ; A119: for x1, x2 being set st x1 in dom k & x2 in dom k & k . x1 = k . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom k & x2 in dom k & k . x1 = k . x2 implies x1 = x2 ) assume that A120: x1 in dom k and A121: x2 in dom k and A122: k . x1 = k . x2 ; ::_thesis: x1 = x2 reconsider r1 = x1 as Point of I[01] by A120; reconsider r2 = x2 as Point of I[01] by A121; A123: k . x1 = h . (g1 . x1) by A120, FUNCT_1:12 .= (h1 . (g1 . r1)) + (h2 . (g1 . r1)) by A99 .= ((g1 . r1) `1) + (proj2 . (g1 . r1)) by PSCOMP_1:def_5 .= ((g1 . r1) `1) + ((g1 . r1) `2) by PSCOMP_1:def_6 ; A124: k . x2 = h . (g1 . x2) by A121, FUNCT_1:12 .= (h1 . (g1 . r2)) + (h2 . (g1 . r2)) by A99 .= ((g1 . r2) `1) + (proj2 . (g1 . r2)) by PSCOMP_1:def_5 .= ((g1 . r2) `1) + ((g1 . r2) `2) by PSCOMP_1:def_6 ; A125: g . r1 in Lower_Arc (rectangle (a,b,c,d)) by A95; A126: g . r2 in Lower_Arc (rectangle (a,b,c,d)) by A95; reconsider gr1 = g . r1 as Point of (TOP-REAL 2) by A125; reconsider gr2 = g . r2 as Point of (TOP-REAL 2) by A126; now__::_thesis:_(_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,d]|,|[b,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,d]|,|[b,c]|)_&_x1_=_x2_)_or_(_g_._r1_in_LSeg_(|[b,c]|,|[a,c]|)_&_g_._r2_in_LSeg_(|[b,c]|,|[a,c]|)_&_x1_=_x2_)_) percases ( ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) or ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ) by A82, A95, XBOOLE_0:def_3; caseA127: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A128: gr1 `1 = b by A2, Th1; gr2 `1 = b by A2, A127, Th1; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A122, A123, A124, A128, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A93, A94, FUNCT_1:def_4; ::_thesis: verum end; caseA129: ( g . r1 in LSeg (|[b,d]|,|[b,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A130: gr1 `1 = b by A2, Th1; A131: c <= gr1 `2 by A2, A129, Th1; A132: gr2 `2 = c by A1, A129, Th3; A133: gr2 `1 <= b by A1, A129, Th3; A134: b + (gr1 `2) = (gr2 `1) + c by A2, A122, A123, A124, A129, A132, Th1; A135: now__::_thesis:_not_b_<>_gr2_`1 assume b <> gr2 `1 ; ::_thesis: contradiction then b > gr2 `1 by A133, XXREAL_0:1; hence contradiction by A122, A123, A124, A130, A131, A132, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr1_`2_<>_c assume gr1 `2 <> c ; ::_thesis: contradiction then c < gr1 `2 by A131, XXREAL_0:1; hence contradiction by A133, A134, XREAL_1:8; ::_thesis: verum end; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A130, A132, A135, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A93, A94, FUNCT_1:def_4; ::_thesis: verum end; caseA136: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,d]|,|[b,c]|) ) ; ::_thesis: x1 = x2 then A137: gr2 `1 = b by A2, Th1; A138: c <= gr2 `2 by A2, A136, Th1; A139: gr1 `2 = c by A1, A136, Th3; A140: gr1 `1 <= b by A1, A136, Th3; A141: b + (gr2 `2) = (gr1 `1) + c by A1, A122, A123, A124, A136, A137, Th3; A142: now__::_thesis:_not_b_<>_gr1_`1 assume b <> gr1 `1 ; ::_thesis: contradiction then b > gr1 `1 by A140, XXREAL_0:1; hence contradiction by A138, A141, XREAL_1:8; ::_thesis: verum end; now__::_thesis:_not_gr2_`2_<>_c assume gr2 `2 <> c ; ::_thesis: contradiction then c < gr2 `2 by A138, XXREAL_0:1; hence contradiction by A122, A123, A124, A137, A139, A140, XREAL_1:8; ::_thesis: verum end; then |[(gr2 `1),(gr2 `2)]| = g . r1 by A137, A139, A142, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A93, A94, FUNCT_1:def_4; ::_thesis: verum end; caseA143: ( g . r1 in LSeg (|[b,c]|,|[a,c]|) & g . r2 in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: x1 = x2 then A144: gr1 `2 = c by A1, Th3; gr2 `2 = c by A1, A143, Th3; then |[(gr1 `1),(gr1 `2)]| = g . r2 by A122, A123, A124, A144, EUCLID:53; then g . r1 = g . r2 by EUCLID:53; hence x1 = x2 by A93, A94, FUNCT_1:def_4; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; A145: dom k = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then s1 in dom k by A88, A89, XXREAL_1:1; then rxc = s1 by A111, A117, A118, A119, A145; hence contradiction by A102, A117, XXREAL_1:1; ::_thesis: verum end; hence s1 <= s2 ; ::_thesis: verum end; then LE p1,p2, Lower_Arc (rectangle (a,b,c,d)), E-max (rectangle (a,b,c,d)), W-min (rectangle (a,b,c,d)) by A83, A84, JORDAN5C:def_3; hence LE p1,p2, rectangle (a,b,c,d) by A80, A83, A84, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p1,p2, rectangle (a,b,c,d) ; ::_thesis: verum end; end; theorem Th63: :: JGRAPH_6:63 for x being set for a, b, c, d being real number st x in rectangle (a,b,c,d) & a < b & c < d & not x in LSeg (|[a,c]|,|[a,d]|) & not x in LSeg (|[a,d]|,|[b,d]|) & not x in LSeg (|[b,d]|,|[b,c]|) holds x in LSeg (|[b,c]|,|[a,c]|) proof let x be set ; ::_thesis: for a, b, c, d being real number st x in rectangle (a,b,c,d) & a < b & c < d & not x in LSeg (|[a,c]|,|[a,d]|) & not x in LSeg (|[a,d]|,|[b,d]|) & not x in LSeg (|[b,d]|,|[b,c]|) holds x in LSeg (|[b,c]|,|[a,c]|) let a, b, c, d be real number ; ::_thesis: ( x in rectangle (a,b,c,d) & a < b & c < d & not x in LSeg (|[a,c]|,|[a,d]|) & not x in LSeg (|[a,d]|,|[b,d]|) & not x in LSeg (|[b,d]|,|[b,c]|) implies x in LSeg (|[b,c]|,|[a,c]|) ) assume that A1: x in rectangle (a,b,c,d) and A2: a < b and A3: c < d ; ::_thesis: ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) x in { q where q is Point of (TOP-REAL 2) : ( ( q `1 = a & q `2 <= d & q `2 >= c ) or ( q `1 <= b & q `1 >= a & q `2 = d ) or ( q `1 <= b & q `1 >= a & q `2 = c ) or ( q `1 = b & q `2 <= d & q `2 >= c ) ) } by A1, A2, A3, SPPOL_2:54; then consider p being Point of (TOP-REAL 2) such that A4: p = x and A5: ( ( p `1 = a & p `2 <= d & p `2 >= c ) or ( p `1 <= b & p `1 >= a & p `2 = d ) or ( p `1 <= b & p `1 >= a & p `2 = c ) or ( p `1 = b & p `2 <= d & p `2 >= c ) ) ; now__::_thesis:_(_(_p_`1_=_a_&_c_<=_p_`2_&_p_`2_<=_d_&_(_x_in_LSeg_(|[a,c]|,|[a,d]|)_or_x_in_LSeg_(|[a,d]|,|[b,d]|)_or_x_in_LSeg_(|[b,d]|,|[b,c]|)_or_x_in_LSeg_(|[b,c]|,|[a,c]|)_)_)_or_(_p_`2_=_d_&_a_<=_p_`1_&_p_`1_<=_b_&_(_x_in_LSeg_(|[a,c]|,|[a,d]|)_or_x_in_LSeg_(|[a,d]|,|[b,d]|)_or_x_in_LSeg_(|[b,d]|,|[b,c]|)_or_x_in_LSeg_(|[b,c]|,|[a,c]|)_)_)_or_(_p_`1_=_b_&_c_<=_p_`2_&_p_`2_<=_d_&_(_x_in_LSeg_(|[a,c]|,|[a,d]|)_or_x_in_LSeg_(|[a,d]|,|[b,d]|)_or_x_in_LSeg_(|[b,d]|,|[b,c]|)_or_x_in_LSeg_(|[b,c]|,|[a,c]|)_)_)_or_(_p_`2_=_c_&_a_<=_p_`1_&_p_`1_<=_b_&_(_x_in_LSeg_(|[a,c]|,|[a,d]|)_or_x_in_LSeg_(|[a,d]|,|[b,d]|)_or_x_in_LSeg_(|[b,d]|,|[b,c]|)_or_x_in_LSeg_(|[b,c]|,|[a,c]|)_)_)_) percases ( ( p `1 = a & c <= p `2 & p `2 <= d ) or ( p `2 = d & a <= p `1 & p `1 <= b ) or ( p `1 = b & c <= p `2 & p `2 <= d ) or ( p `2 = c & a <= p `1 & p `1 <= b ) ) by A5; caseA6: ( p `1 = a & c <= p `2 & p `2 <= d ) ; ::_thesis: ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) A7: d - c > 0 by A3, XREAL_1:50; A8: (p `2) - c >= 0 by A6, XREAL_1:48; A9: d - (p `2) >= 0 by A6, XREAL_1:48; reconsider r = ((p `2) - c) / (d - c) as Real ; A10: 1 - r = ((d - c) / (d - c)) - (((p `2) - c) / (d - c)) by A7, XCMPLX_1:60 .= ((d - c) - ((p `2) - c)) / (d - c) by XCMPLX_1:120 .= (d - (p `2)) / (d - c) ; then A11: (1 - r) + r >= 0 + r by A7, A9, XREAL_1:7; A12: (((1 - r) * |[a,c]|) + (r * |[a,d]|)) `1 = (((1 - r) * |[a,c]|) `1) + ((r * |[a,d]|) `1) by TOPREAL3:2 .= ((1 - r) * (|[a,c]| `1)) + ((r * |[a,d]|) `1) by TOPREAL3:4 .= ((1 - r) * a) + ((r * |[a,d]|) `1) by EUCLID:52 .= ((1 - r) * a) + (r * (|[a,d]| `1)) by TOPREAL3:4 .= ((1 - r) * a) + (r * a) by EUCLID:52 .= p `1 by A6 ; (((1 - r) * |[a,c]|) + (r * |[a,d]|)) `2 = (((1 - r) * |[a,c]|) `2) + ((r * |[a,d]|) `2) by TOPREAL3:2 .= ((1 - r) * (|[a,c]| `2)) + ((r * |[a,d]|) `2) by TOPREAL3:4 .= ((1 - r) * c) + ((r * |[a,d]|) `2) by EUCLID:52 .= ((1 - r) * c) + (r * (|[a,d]| `2)) by TOPREAL3:4 .= (((d - (p `2)) / (d - c)) * c) + ((((p `2) - c) / (d - c)) * d) by A10, EUCLID:52 .= (((d - (p `2)) * ((d - c) ")) * c) + ((((p `2) - c) / (d - c)) * d) by XCMPLX_0:def_9 .= (((d - c) ") * ((d - (p `2)) * c)) + ((((d - c) ") * ((p `2) - c)) * d) by XCMPLX_0:def_9 .= (((d - c) ") * (d - c)) * (p `2) .= 1 * (p `2) by A7, XCMPLX_0:def_7 .= p `2 ; then p = |[((((1 - r) * |[a,c]|) + (r * |[a,d]|)) `1),((((1 - r) * |[a,c]|) + (r * |[a,d]|)) `2)]| by A12, EUCLID:53 .= ((1 - r) * |[a,c]|) + (r * |[a,d]|) by EUCLID:53 ; hence ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) by A4, A7, A8, A11; ::_thesis: verum end; caseA13: ( p `2 = d & a <= p `1 & p `1 <= b ) ; ::_thesis: ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) A14: b - a > 0 by A2, XREAL_1:50; A15: (p `1) - a >= 0 by A13, XREAL_1:48; A16: b - (p `1) >= 0 by A13, XREAL_1:48; reconsider r = ((p `1) - a) / (b - a) as Real ; A17: 1 - r = ((b - a) / (b - a)) - (((p `1) - a) / (b - a)) by A14, XCMPLX_1:60 .= ((b - a) - ((p `1) - a)) / (b - a) by XCMPLX_1:120 .= (b - (p `1)) / (b - a) ; then A18: (1 - r) + r >= 0 + r by A14, A16, XREAL_1:7; A19: (((1 - r) * |[a,d]|) + (r * |[b,d]|)) `1 = (((1 - r) * |[a,d]|) `1) + ((r * |[b,d]|) `1) by TOPREAL3:2 .= ((1 - r) * (|[a,d]| `1)) + ((r * |[b,d]|) `1) by TOPREAL3:4 .= ((1 - r) * a) + ((r * |[b,d]|) `1) by EUCLID:52 .= ((1 - r) * a) + (r * (|[b,d]| `1)) by TOPREAL3:4 .= (((b - (p `1)) / (b - a)) * a) + ((((p `1) - a) / (b - a)) * b) by A17, EUCLID:52 .= (((b - (p `1)) * ((b - a) ")) * a) + ((((p `1) - a) / (b - a)) * b) by XCMPLX_0:def_9 .= (((b - a) ") * ((b - (p `1)) * a)) + ((((b - a) ") * ((p `1) - a)) * b) by XCMPLX_0:def_9 .= (((b - a) ") * (b - a)) * (p `1) .= 1 * (p `1) by A14, XCMPLX_0:def_7 .= p `1 ; (((1 - r) * |[a,d]|) + (r * |[b,d]|)) `2 = (((1 - r) * |[a,d]|) `2) + ((r * |[b,d]|) `2) by TOPREAL3:2 .= ((1 - r) * (|[a,d]| `2)) + ((r * |[b,d]|) `2) by TOPREAL3:4 .= ((1 - r) * d) + ((r * |[b,d]|) `2) by EUCLID:52 .= ((1 - r) * d) + (r * (|[b,d]| `2)) by TOPREAL3:4 .= ((1 - r) * d) + (r * d) by EUCLID:52 .= p `2 by A13 ; then p = |[((((1 - r) * |[a,d]|) + (r * |[b,d]|)) `1),((((1 - r) * |[a,d]|) + (r * |[b,d]|)) `2)]| by A19, EUCLID:53 .= ((1 - r) * |[a,d]|) + (r * |[b,d]|) by EUCLID:53 ; hence ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) by A4, A14, A15, A18; ::_thesis: verum end; caseA20: ( p `1 = b & c <= p `2 & p `2 <= d ) ; ::_thesis: ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) A21: d - c > 0 by A3, XREAL_1:50; A22: (p `2) - c >= 0 by A20, XREAL_1:48; A23: d - (p `2) >= 0 by A20, XREAL_1:48; reconsider r = (d - (p `2)) / (d - c) as Real ; A24: 1 - r = ((d - c) / (d - c)) - ((d - (p `2)) / (d - c)) by A21, XCMPLX_1:60 .= ((d - c) - (d - (p `2))) / (d - c) by XCMPLX_1:120 .= ((p `2) - c) / (d - c) ; then A25: (1 - r) + r >= 0 + r by A21, A22, XREAL_1:7; A26: (((1 - r) * |[b,d]|) + (r * |[b,c]|)) `1 = (((1 - r) * |[b,d]|) `1) + ((r * |[b,c]|) `1) by TOPREAL3:2 .= ((1 - r) * (|[b,d]| `1)) + ((r * |[b,c]|) `1) by TOPREAL3:4 .= ((1 - r) * b) + ((r * |[b,c]|) `1) by EUCLID:52 .= ((1 - r) * b) + (r * (|[b,c]| `1)) by TOPREAL3:4 .= ((1 - r) * b) + (r * b) by EUCLID:52 .= p `1 by A20 ; (((1 - r) * |[b,d]|) + (r * |[b,c]|)) `2 = (((1 - r) * |[b,d]|) `2) + ((r * |[b,c]|) `2) by TOPREAL3:2 .= ((1 - r) * (|[b,d]| `2)) + ((r * |[b,c]|) `2) by TOPREAL3:4 .= ((1 - r) * d) + ((r * |[b,c]|) `2) by EUCLID:52 .= ((1 - r) * d) + (r * (|[b,c]| `2)) by TOPREAL3:4 .= ((((p `2) - c) / (d - c)) * d) + (((d - (p `2)) / (d - c)) * c) by A24, EUCLID:52 .= ((((p `2) - c) * ((d - c) ")) * d) + (((d - (p `2)) / (d - c)) * c) by XCMPLX_0:def_9 .= (((d - c) ") * (((p `2) - c) * d)) + ((((d - c) ") * (d - (p `2))) * c) by XCMPLX_0:def_9 .= (((d - c) ") * (d - c)) * (p `2) .= 1 * (p `2) by A21, XCMPLX_0:def_7 .= p `2 ; then p = |[((((1 - r) * |[b,d]|) + (r * |[b,c]|)) `1),((((1 - r) * |[b,d]|) + (r * |[b,c]|)) `2)]| by A26, EUCLID:53 .= ((1 - r) * |[b,d]|) + (r * |[b,c]|) by EUCLID:53 ; hence ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) by A4, A21, A23, A25; ::_thesis: verum end; caseA27: ( p `2 = c & a <= p `1 & p `1 <= b ) ; ::_thesis: ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) A28: b - a > 0 by A2, XREAL_1:50; A29: (p `1) - a >= 0 by A27, XREAL_1:48; A30: b - (p `1) >= 0 by A27, XREAL_1:48; reconsider r = (b - (p `1)) / (b - a) as Real ; A31: 1 - r = ((b - a) / (b - a)) - ((b - (p `1)) / (b - a)) by A28, XCMPLX_1:60 .= ((b - a) - (b - (p `1))) / (b - a) by XCMPLX_1:120 .= ((p `1) - a) / (b - a) ; then A32: (1 - r) + r >= 0 + r by A28, A29, XREAL_1:7; A33: (((1 - r) * |[b,c]|) + (r * |[a,c]|)) `1 = (((1 - r) * |[b,c]|) `1) + ((r * |[a,c]|) `1) by TOPREAL3:2 .= ((1 - r) * (|[b,c]| `1)) + ((r * |[a,c]|) `1) by TOPREAL3:4 .= ((1 - r) * b) + ((r * |[a,c]|) `1) by EUCLID:52 .= ((1 - r) * b) + (r * (|[a,c]| `1)) by TOPREAL3:4 .= ((((p `1) - a) / (b - a)) * b) + (((b - (p `1)) / (b - a)) * a) by A31, EUCLID:52 .= ((((p `1) - a) * ((b - a) ")) * b) + (((b - (p `1)) / (b - a)) * a) by XCMPLX_0:def_9 .= (((b - a) ") * (((p `1) - a) * b)) + ((((b - a) ") * (b - (p `1))) * a) by XCMPLX_0:def_9 .= (((b - a) ") * (b - a)) * (p `1) .= 1 * (p `1) by A28, XCMPLX_0:def_7 .= p `1 ; (((1 - r) * |[b,c]|) + (r * |[a,c]|)) `2 = (((1 - r) * |[b,c]|) `2) + ((r * |[a,c]|) `2) by TOPREAL3:2 .= ((1 - r) * (|[b,c]| `2)) + ((r * |[a,c]|) `2) by TOPREAL3:4 .= ((1 - r) * c) + ((r * |[a,c]|) `2) by EUCLID:52 .= ((1 - r) * c) + (r * (|[a,c]| `2)) by TOPREAL3:4 .= ((1 - r) * c) + (r * c) by EUCLID:52 .= p `2 by A27 ; then p = |[((((1 - r) * |[b,c]|) + (r * |[a,c]|)) `1),((((1 - r) * |[b,c]|) + (r * |[a,c]|)) `2)]| by A33, EUCLID:53 .= ((1 - r) * |[b,c]|) + (r * |[a,c]|) by EUCLID:53 ; hence ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) by A4, A28, A30, A32; ::_thesis: verum end; end; end; hence ( x in LSeg (|[a,c]|,|[a,d]|) or x in LSeg (|[a,d]|,|[b,d]|) or x in LSeg (|[b,d]|,|[b,c]|) or x in LSeg (|[b,c]|,|[a,c]|) ) ; ::_thesis: verum end; begin theorem Th64: :: JGRAPH_6:64 for p1, p2 being Point of (TOP-REAL 2) st LE p1,p2, rectangle ((- 1),1,(- 1),1) & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & not ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p2 `2 >= p1 `2 ) & not p2 in LSeg (|[(- 1),1]|,|[1,1]|) & not p2 in LSeg (|[1,1]|,|[1,(- 1)]|) holds ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> |[(- 1),(- 1)]| ) proof let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( LE p1,p2, rectangle ((- 1),1,(- 1),1) & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & not ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p2 `2 >= p1 `2 ) & not p2 in LSeg (|[(- 1),1]|,|[1,1]|) & not p2 in LSeg (|[1,1]|,|[1,(- 1)]|) implies ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> |[(- 1),(- 1)]| ) ) set K = rectangle ((- 1),1,(- 1),1); assume that A1: LE p1,p2, rectangle ((- 1),1,(- 1),1) and A2: p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ; ::_thesis: ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p2 `2 >= p1 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> |[(- 1),(- 1)]| ) ) ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A1, A2, Th59; hence ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p2 `2 >= p1 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> |[(- 1),(- 1)]| ) ) by Th46; ::_thesis: verum end; theorem Th65: :: JGRAPH_6:65 for p1, p2 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 >= 0 & LE p1,p2, rectangle ((- 1),1,(- 1),1) holds LE f . p1,f . p2,P proof let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 >= 0 & LE p1,p2, rectangle ((- 1),1,(- 1),1) holds LE f . p1,f . p2,P let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 >= 0 & LE p1,p2, rectangle ((- 1),1,(- 1),1) holds LE f . p1,f . p2,P let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( P = circle (0,0,1) & f = Sq_Circ & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 >= 0 & LE p1,p2, rectangle ((- 1),1,(- 1),1) implies LE f . p1,f . p2,P ) set K = rectangle ((- 1),1,(- 1),1); assume that A1: P = circle (0,0,1) and A2: f = Sq_Circ and A3: p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) and A4: p1 `2 >= 0 and A5: LE p1,p2, rectangle ((- 1),1,(- 1),1) ; ::_thesis: LE f . p1,f . p2,P A6: rectangle ((- 1),1,(- 1),1) is being_simple_closed_curve by Th50; A7: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } by A1, Th24; A8: p1 `1 = - 1 by A3, Th1; A9: p1 `2 <= 1 by A3, Th1; A10: p1 in rectangle ((- 1),1,(- 1),1) by A5, A6, JORDAN7:5; A11: p2 in rectangle ((- 1),1,(- 1),1) by A5, A6, JORDAN7:5; A12: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A7, Lm15, Th35, JGRAPH_3:23; A13: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A14: f . p1 in P by A10, A12, FUNCT_1:def_6; A15: f . p2 in P by A11, A12, A13, FUNCT_1:def_6; A16: p1 `1 = - 1 by A3, Th1; A17: (p1 `2) ^2 >= 0 by XREAL_1:63; then A18: sqrt (1 + ((p1 `2) ^2)) > 0 by SQUARE_1:25; A19: p1 `2 <= - (p1 `1) by A3, A8, Th1; p1 <> 0. (TOP-REAL 2) by A8, EUCLID:52, EUCLID:54; then A20: f . p1 = |[((p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| by A2, A4, A16, A19, JGRAPH_3:def_1; then A21: (f . p1) `1 = (p1 `1) / (sqrt (1 + (((p1 `2) / (- 1)) ^2))) by A16, EUCLID:52 .= (p1 `1) / (sqrt (1 + ((p1 `2) ^2))) ; A22: (f . p1) `2 = (p1 `2) / (sqrt (1 + (((p1 `2) / (- 1)) ^2))) by A16, A20, EUCLID:52 .= (p1 `2) / (sqrt (1 + ((p1 `2) ^2))) ; A23: (f . p1) `1 < 0 by A16, A17, A21, SQUARE_1:25, XREAL_1:141; A24: (f . p1) `2 >= 0 by A4, A18, A22; f . p1 in { p9 where p9 is Point of (TOP-REAL 2) : ( p9 in P & p9 `2 >= 0 ) } by A4, A14, A18, A22; then A25: f . p1 in Upper_Arc P by A7, JGRAPH_5:34; now__::_thesis:_(_(_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_p2_`2_>=_p1_`2_&_LE_f_._p1,f_._p2,P_)_or_(_p2_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_LE_f_._p1,f_._p2,P_)_or_(_p2_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_LE_f_._p1,f_._p2,P_)_or_(_p2_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p2_<>_|[(-_1),(-_1)]|_&_LE_f_._p1,f_._p2,P_)_) percases ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p2 `2 >= p1 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> |[(- 1),(- 1)]| ) ) by A3, A5, Th64; caseA26: ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p2 `2 >= p1 `2 ) ; ::_thesis: LE f . p1,f . p2,P A27: (p2 `2) ^2 >= 0 by XREAL_1:63; then A28: sqrt (1 + ((p2 `2) ^2)) > 0 by SQUARE_1:25; A29: p2 `1 = - 1 by A26, Th1; A30: - 1 <= p2 `2 by A26, Th1; A31: p2 `2 <= - (p2 `1) by A26, A29, Th1; p2 <> 0. (TOP-REAL 2) by A29, EUCLID:52, EUCLID:54; then A32: f . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A2, A29, A30, A31, JGRAPH_3:def_1; then A33: (f . p2) `1 = (p2 `1) / (sqrt (1 + (((p2 `2) / (- 1)) ^2))) by A29, EUCLID:52 .= (p2 `1) / (sqrt (1 + ((p2 `2) ^2))) ; A34: (f . p2) `2 = (p2 `2) / (sqrt (1 + (((p2 `2) / (- 1)) ^2))) by A29, A32, EUCLID:52 .= (p2 `2) / (sqrt (1 + ((p2 `2) ^2))) ; A35: (f . p2) `1 < 0 by A27, A29, A33, SQUARE_1:25, XREAL_1:141; (p1 `2) * (sqrt (1 + ((p2 `2) ^2))) <= (p2 `2) * (sqrt (1 + ((p1 `2) ^2))) by A4, A26, Lm3; then ((p1 `2) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2))) <= ((p2 `2) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2))) by A28, XREAL_1:72; then p1 `2 <= ((p2 `2) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2))) by A28, XCMPLX_1:89; then (p1 `2) / (sqrt (1 + ((p1 `2) ^2))) <= (((p2 `2) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p1 `2) ^2))) by A18, XREAL_1:72; then (p1 `2) / (sqrt (1 + ((p1 `2) ^2))) <= (((p2 `2) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2))) by XCMPLX_1:48; then (f . p1) `2 <= (f . p2) `2 by A18, A22, A34, XCMPLX_1:89; hence LE f . p1,f . p2,P by A7, A14, A15, A23, A24, A35, JGRAPH_5:53; ::_thesis: verum end; caseA36: p2 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: LE f . p1,f . p2,P then A37: p2 `2 = 1 by Th3; A38: - 1 <= p2 `1 by A36, Th3; A39: p2 `1 <= 1 by A36, Th3; (p2 `1) ^2 >= 0 by XREAL_1:63; then A40: sqrt (1 + ((p2 `1) ^2)) > 0 by SQUARE_1:25; p2 <> 0. (TOP-REAL 2) by A37, EUCLID:52, EUCLID:54; then A41: f . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| by A2, A37, A38, A39, JGRAPH_3:4; then A42: (f . p2) `1 = (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) by A37, EUCLID:52; A43: (f . p2) `2 >= 0 by A37, A40, A41, EUCLID:52; - (sqrt (1 + ((p2 `1) ^2))) <= (p2 `1) * (sqrt (1 + ((p1 `2) ^2))) by A4, A9, A38, A39, SQUARE_1:55; then ((p1 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p2 `1) ^2))) <= ((p2 `1) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `1) ^2))) by A8, A40, XREAL_1:72; then p1 `1 <= ((p2 `1) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `1) ^2))) by A40, XCMPLX_1:89; then (p1 `1) / (sqrt (1 + ((p1 `2) ^2))) <= (((p2 `1) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p1 `2) ^2))) by A18, XREAL_1:72; then (p1 `1) / (sqrt (1 + ((p1 `2) ^2))) <= (((p2 `1) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `1) ^2))) by XCMPLX_1:48; then (f . p1) `1 <= (f . p2) `1 by A18, A21, A42, XCMPLX_1:89; hence LE f . p1,f . p2,P by A4, A7, A14, A15, A18, A22, A43, JGRAPH_5:54; ::_thesis: verum end; caseA44: p2 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: LE f . p1,f . p2,P then A45: p2 `1 = 1 by Th1; A46: - 1 <= p2 `2 by A44, Th1; A47: p2 `2 <= 1 by A44, Th1; (p2 `2) ^2 >= 0 by XREAL_1:63; then A48: sqrt (1 + ((p2 `2) ^2)) > 0 by SQUARE_1:25; p2 <> 0. (TOP-REAL 2) by A45, EUCLID:52, EUCLID:54; then f . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A2, A45, A46, A47, JGRAPH_3:def_1; then A49: (f . p2) `1 = (p2 `1) / (sqrt (1 + ((p2 `2) ^2))) by A45, EUCLID:52; (p1 `1) / (sqrt (1 + ((p1 `2) ^2))) <= (((p2 `1) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p1 `2) ^2))) by A8, A18, A45, A48, XREAL_1:72; then (p1 `1) / (sqrt (1 + ((p1 `2) ^2))) <= (((p2 `1) * (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p1 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2))) by XCMPLX_1:48; then A50: (f . p1) `1 <= (f . p2) `1 by A18, A21, A49, XCMPLX_1:89; now__::_thesis:_(_(_(f_._p2)_`2_>=_0_&_LE_f_._p1,f_._p2,P_)_or_(_(f_._p2)_`2_<_0_&_LE_f_._p1,f_._p2,P_)_) percases ( (f . p2) `2 >= 0 or (f . p2) `2 < 0 ) ; case (f . p2) `2 >= 0 ; ::_thesis: LE f . p1,f . p2,P hence LE f . p1,f . p2,P by A4, A7, A14, A15, A18, A22, A50, JGRAPH_5:54; ::_thesis: verum end; caseA51: (f . p2) `2 < 0 ; ::_thesis: LE f . p1,f . p2,P then f . p2 in { p9 where p9 is Point of (TOP-REAL 2) : ( p9 in P & p9 `2 <= 0 ) } by A15; then A52: f . p2 in Lower_Arc P by A7, JGRAPH_5:35; W-min P = |[(- 1),0]| by A7, JGRAPH_5:29; then f . p2 <> W-min P by A51, EUCLID:52; hence LE f . p1,f . p2,P by A25, A52, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE f . p1,f . p2,P ; ::_thesis: verum end; caseA53: ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> |[(- 1),(- 1)]| ) ; ::_thesis: LE f . p1,f . p2,P then A54: p2 `2 = - 1 by Th3; A55: - 1 <= p2 `1 by A53, Th3; A56: (p2 `1) ^2 >= 0 by XREAL_1:63; A57: p2 `1 <= - (p2 `2) by A53, A54, Th3; p2 <> 0. (TOP-REAL 2) by A54, EUCLID:52, EUCLID:54; then f . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| by A2, A54, A55, A57, JGRAPH_3:4; then (f . p2) `2 = (p2 `2) / (sqrt (1 + (((p2 `1) / (- 1)) ^2))) by A54, EUCLID:52 .= (p2 `2) / (sqrt (1 + ((p2 `1) ^2))) ; then A58: (f . p2) `2 < 0 by A54, A56, SQUARE_1:25, XREAL_1:141; then f . p2 in { p9 where p9 is Point of (TOP-REAL 2) : ( p9 in P & p9 `2 <= 0 ) } by A15; then A59: f . p2 in Lower_Arc P by A7, JGRAPH_5:35; W-min P = |[(- 1),0]| by A7, JGRAPH_5:29; then f . p2 <> W-min P by A58, EUCLID:52; hence LE f . p1,f . p2,P by A25, A59, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE f . p1,f . p2,P ; ::_thesis: verum end; theorem Th66: :: JGRAPH_6:66 for p1, p2, p3 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 >= 0 & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) holds LE f . p2,f . p3,P proof let p1, p2, p3 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 >= 0 & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) holds LE f . p2,f . p3,P let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 >= 0 & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) holds LE f . p2,f . p3,P let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( P = circle (0,0,1) & f = Sq_Circ & p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p1 `2 >= 0 & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) implies LE f . p2,f . p3,P ) set K = rectangle ((- 1),1,(- 1),1); assume that A1: P = circle (0,0,1) and A2: f = Sq_Circ and A3: p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) and A4: p1 `2 >= 0 and A5: LE p1,p2, rectangle ((- 1),1,(- 1),1) and A6: LE p2,p3, rectangle ((- 1),1,(- 1),1) ; ::_thesis: LE f . p2,f . p3,P A7: rectangle ((- 1),1,(- 1),1) is being_simple_closed_curve by Th50; A8: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } by A1, Th24; A9: p3 in rectangle ((- 1),1,(- 1),1) by A6, A7, JORDAN7:5; A10: p2 in rectangle ((- 1),1,(- 1),1) by A5, A7, JORDAN7:5; A11: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A8, Lm15, Th35, JGRAPH_3:23; A12: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A13: f . p2 in P by A10, A11, FUNCT_1:def_6; A14: f . p3 in P by A9, A11, A12, FUNCT_1:def_6; now__::_thesis:_(_(_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_p2_`2_>=_p1_`2_&_LE_f_._p2,f_._p3,P_)_or_(_p2_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_LE_f_._p2,f_._p3,P_)_or_(_p2_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_LE_f_._p2,f_._p3,P_)_or_(_p2_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p2_<>_|[(-_1),(-_1)]|_&_LE_f_._p2,f_._p3,P_)_) percases ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p2 `2 >= p1 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> |[(- 1),(- 1)]| ) ) by A3, A5, Th64; case ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p2 `2 >= p1 `2 ) ; ::_thesis: LE f . p2,f . p3,P hence LE f . p2,f . p3,P by A1, A2, A4, A6, Th65; ::_thesis: verum end; caseA15: p2 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: LE f . p2,f . p3,P then A16: p2 `2 = 1 by Th3; A17: - 1 <= p2 `1 by A15, Th3; A18: p2 `1 <= 1 by A15, Th3; (p2 `1) ^2 >= 0 by XREAL_1:63; then A19: sqrt (1 + ((p2 `1) ^2)) > 0 by SQUARE_1:25; p2 <> 0. (TOP-REAL 2) by A16, EUCLID:52, EUCLID:54; then A20: f . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| by A2, A16, A17, A18, JGRAPH_3:4; then A21: (f . p2) `1 = (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) by A16, EUCLID:52; A22: (f . p2) `2 >= 0 by A16, A19, A20, EUCLID:52; then f . p2 in { p9 where p9 is Point of (TOP-REAL 2) : ( p9 in P & p9 `2 >= 0 ) } by A13; then A23: f . p2 in Upper_Arc P by A8, JGRAPH_5:34; now__::_thesis:_(_(_p3_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_p2_`1_<=_p3_`1_&_LE_f_._p2,f_._p3,P_)_or_(_p3_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_LE_f_._p2,f_._p3,P_)_or_(_p3_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p3_<>_W-min_(rectangle_((-_1),1,(-_1),1))_&_LE_f_._p2,f_._p3,P_)_) percases ( ( p3 in LSeg (|[(- 1),1]|,|[1,1]|) & p2 `1 <= p3 `1 ) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A6, A15, Th60; caseA24: ( p3 in LSeg (|[(- 1),1]|,|[1,1]|) & p2 `1 <= p3 `1 ) ; ::_thesis: LE f . p2,f . p3,P then A25: p3 `2 = 1 by Th3; A26: - 1 <= p3 `1 by A24, Th3; A27: p3 `1 <= 1 by A24, Th3; (p3 `1) ^2 >= 0 by XREAL_1:63; then A28: sqrt (1 + ((p3 `1) ^2)) > 0 by SQUARE_1:25; p3 <> 0. (TOP-REAL 2) by A25, EUCLID:52, EUCLID:54; then A29: f . p3 = |[((p3 `1) / (sqrt (1 + (((p3 `1) / (p3 `2)) ^2)))),((p3 `2) / (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))))]| by A2, A25, A26, A27, JGRAPH_3:4; then A30: (f . p3) `1 = (p3 `1) / (sqrt (1 + ((p3 `1) ^2))) by A25, EUCLID:52; A31: (f . p3) `2 >= 0 by A25, A28, A29, EUCLID:52; (p2 `1) * (sqrt (1 + ((p3 `1) ^2))) <= (p3 `1) * (sqrt (1 + ((p2 `1) ^2))) by A24, SQUARE_1:57; then ((p2 `1) * (sqrt (1 + ((p3 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) <= ((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by A28, XREAL_1:72; then p2 `1 <= ((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by A28, XCMPLX_1:89; then (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) <= (((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2)))) / (sqrt (1 + ((p2 `1) ^2))) by A19, XREAL_1:72; then (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) <= (((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by XCMPLX_1:48; then (f . p2) `1 <= (f . p3) `1 by A19, A21, A30, XCMPLX_1:89; hence LE f . p2,f . p3,P by A8, A13, A14, A22, A31, JGRAPH_5:54; ::_thesis: verum end; caseA32: p3 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: LE f . p2,f . p3,P then A33: p3 `1 = 1 by Th1; A34: - 1 <= p3 `2 by A32, Th1; A35: p3 `2 <= 1 by A32, Th1; (p3 `2) ^2 >= 0 by XREAL_1:63; then A36: sqrt (1 + ((p3 `2) ^2)) > 0 by SQUARE_1:25; p3 <> 0. (TOP-REAL 2) by A33, EUCLID:52, EUCLID:54; then f . p3 = |[((p3 `1) / (sqrt (1 + (((p3 `2) / (p3 `1)) ^2)))),((p3 `2) / (sqrt (1 + (((p3 `2) / (p3 `1)) ^2))))]| by A2, A33, A34, A35, JGRAPH_3:def_1; then A37: (f . p3) `1 = (p3 `1) / (sqrt (1 + ((p3 `2) ^2))) by A33, EUCLID:52; A38: - 1 <= - (p2 `1) by A18, XREAL_1:24; A39: - (- 1) >= - (p2 `1) by A17, XREAL_1:24; (p2 `1) ^2 = (- (p2 `1)) ^2 ; then (- (- (p2 `1))) * (sqrt (1 + ((p3 `2) ^2))) <= sqrt (1 + ((p2 `1) ^2)) by A34, A35, A38, A39, SQUARE_1:55; then ((p2 `1) * (sqrt (1 + ((p3 `2) ^2)))) / (sqrt (1 + ((p3 `2) ^2))) <= ((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `2) ^2))) by A33, A36, XREAL_1:72; then p2 `1 <= ((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `2) ^2))) by A36, XCMPLX_1:89; then (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) <= (((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `2) ^2)))) / (sqrt (1 + ((p2 `1) ^2))) by A19, XREAL_1:72; then (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) <= (((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `2) ^2))) by XCMPLX_1:48; then A40: (f . p2) `1 <= (f . p3) `1 by A19, A21, A37, XCMPLX_1:89; now__::_thesis:_(_(_(f_._p3)_`2_>=_0_&_LE_f_._p2,f_._p3,P_)_or_(_(f_._p3)_`2_<_0_&_LE_f_._p2,f_._p3,P_)_) percases ( (f . p3) `2 >= 0 or (f . p3) `2 < 0 ) ; case (f . p3) `2 >= 0 ; ::_thesis: LE f . p2,f . p3,P hence LE f . p2,f . p3,P by A8, A13, A14, A22, A40, JGRAPH_5:54; ::_thesis: verum end; caseA41: (f . p3) `2 < 0 ; ::_thesis: LE f . p2,f . p3,P then f . p3 in { p9 where p9 is Point of (TOP-REAL 2) : ( p9 in P & p9 `2 <= 0 ) } by A14; then A42: f . p3 in Lower_Arc P by A8, JGRAPH_5:35; W-min P = |[(- 1),0]| by A8, JGRAPH_5:29; then f . p3 <> W-min P by A41, EUCLID:52; hence LE f . p2,f . p3,P by A23, A42, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE f . p2,f . p3,P ; ::_thesis: verum end; caseA43: ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ; ::_thesis: LE f . p2,f . p3,P then A44: p3 `2 = - 1 by Th3; A45: - 1 <= p3 `1 by A43, Th3; A46: (p3 `1) ^2 >= 0 by XREAL_1:63; A47: - (p3 `2) >= p3 `1 by A43, A44, Th3; p3 <> 0. (TOP-REAL 2) by A44, EUCLID:52, EUCLID:54; then f . p3 = |[((p3 `1) / (sqrt (1 + (((p3 `1) / (p3 `2)) ^2)))),((p3 `2) / (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))))]| by A2, A44, A45, A47, JGRAPH_3:4; then (f . p3) `2 = (p3 `2) / (sqrt (1 + (((p3 `1) / (- 1)) ^2))) by A44, EUCLID:52 .= (p3 `2) / (sqrt (1 + ((p3 `1) ^2))) ; then A48: (f . p3) `2 < 0 by A44, A46, SQUARE_1:25, XREAL_1:141; then f . p3 in { p9 where p9 is Point of (TOP-REAL 2) : ( p9 in P & p9 `2 <= 0 ) } by A14; then A49: f . p3 in Lower_Arc P by A8, JGRAPH_5:35; W-min P = |[(- 1),0]| by A8, JGRAPH_5:29; then f . p3 <> W-min P by A48, EUCLID:52; hence LE f . p2,f . p3,P by A23, A49, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE f . p2,f . p3,P ; ::_thesis: verum end; caseA50: p2 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: LE f . p2,f . p3,P then A51: p2 `1 = 1 by Th1; A52: - 1 <= p2 `2 by A50, Th1; A53: p2 `2 <= 1 by A50, Th1; (p2 `2) ^2 >= 0 by XREAL_1:63; then A54: sqrt (1 + ((p2 `2) ^2)) > 0 by SQUARE_1:25; p2 <> 0. (TOP-REAL 2) by A51, EUCLID:52, EUCLID:54; then A55: f . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A2, A51, A52, A53, JGRAPH_3:def_1; then A56: (f . p2) `1 = (p2 `1) / (sqrt (1 + ((p2 `2) ^2))) by A51, EUCLID:52; A57: (f . p2) `2 = (p2 `2) / (sqrt (1 + ((p2 `2) ^2))) by A51, A55, EUCLID:52; now__::_thesis:_(_(_p3_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_p2_`2_>=_p3_`2_&_LE_f_._p2,f_._p3,P_)_or_(_p3_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p3_<>_W-min_(rectangle_((-_1),1,(-_1),1))_&_LE_f_._p2,f_._p3,P_)_) percases ( ( p3 in LSeg (|[1,1]|,|[1,(- 1)]|) & p2 `2 >= p3 `2 ) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A6, A50, Th61; caseA58: ( p3 in LSeg (|[1,1]|,|[1,(- 1)]|) & p2 `2 >= p3 `2 ) ; ::_thesis: LE f . p2,f . p3,P then A59: p3 `1 = 1 by Th1; A60: - 1 <= p3 `2 by A58, Th1; A61: p3 `2 <= 1 by A58, Th1; (p3 `2) ^2 >= 0 by XREAL_1:63; then A62: sqrt (1 + ((p3 `2) ^2)) > 0 by SQUARE_1:25; p3 <> 0. (TOP-REAL 2) by A59, EUCLID:52, EUCLID:54; then A63: f . p3 = |[((p3 `1) / (sqrt (1 + (((p3 `2) / (p3 `1)) ^2)))),((p3 `2) / (sqrt (1 + (((p3 `2) / (p3 `1)) ^2))))]| by A2, A59, A60, A61, JGRAPH_3:def_1; then A64: (f . p3) `2 = (p3 `2) / (sqrt (1 + ((p3 `2) ^2))) by A59, EUCLID:52; A65: (f . p3) `1 >= 0 by A59, A62, A63, EUCLID:52; (p2 `2) * (sqrt (1 + ((p3 `2) ^2))) >= (p3 `2) * (sqrt (1 + ((p2 `2) ^2))) by A58, SQUARE_1:57; then ((p2 `2) * (sqrt (1 + ((p3 `2) ^2)))) / (sqrt (1 + ((p3 `2) ^2))) >= ((p3 `2) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p3 `2) ^2))) by A62, XREAL_1:72; then p2 `2 >= ((p3 `2) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p3 `2) ^2))) by A62, XCMPLX_1:89; then (p2 `2) / (sqrt (1 + ((p2 `2) ^2))) >= (((p3 `2) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p3 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2))) by A54, XREAL_1:72; then (p2 `2) / (sqrt (1 + ((p2 `2) ^2))) >= (((p3 `2) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p3 `2) ^2))) by XCMPLX_1:48; then (p2 `2) / (sqrt (1 + ((p2 `2) ^2))) >= (p3 `2) / (sqrt (1 + ((p3 `2) ^2))) by A54, XCMPLX_1:89; hence LE f . p2,f . p3,P by A8, A13, A14, A51, A54, A56, A57, A64, A65, JGRAPH_5:55; ::_thesis: verum end; caseA66: ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ; ::_thesis: LE f . p2,f . p3,P then A67: p3 `2 = - 1 by Th3; A68: - 1 <= p3 `1 by A66, Th3; A69: p3 `1 <= 1 by A66, Th3; A70: (p3 `1) ^2 >= 0 by XREAL_1:63; then A71: sqrt (1 + ((p3 `1) ^2)) > 0 by SQUARE_1:25; A72: - (p3 `2) >= p3 `1 by A66, A67, Th3; p3 <> 0. (TOP-REAL 2) by A67, EUCLID:52, EUCLID:54; then A73: f . p3 = |[((p3 `1) / (sqrt (1 + (((p3 `1) / (p3 `2)) ^2)))),((p3 `2) / (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))))]| by A2, A67, A68, A72, JGRAPH_3:4; then A74: (f . p3) `1 = (p3 `1) / (sqrt (1 + (((p3 `1) / (- 1)) ^2))) by A67, EUCLID:52 .= (p3 `1) / (sqrt (1 + ((p3 `1) ^2))) ; A75: (f . p3) `2 = (p3 `2) / (sqrt (1 + (((p3 `1) / (- 1)) ^2))) by A67, A73, EUCLID:52 .= (p3 `2) / (sqrt (1 + ((p3 `1) ^2))) ; then A76: (f . p3) `2 < 0 by A67, A70, SQUARE_1:25, XREAL_1:141; f . p3 in { p9 where p9 is Point of (TOP-REAL 2) : ( p9 in P & p9 `2 <= 0 ) } by A14, A67, A71, A75; then A77: f . p3 in Lower_Arc P by A8, JGRAPH_5:35; W-min P = |[(- 1),0]| by A8, JGRAPH_5:29; then A78: f . p3 <> W-min P by A76, EUCLID:52; now__::_thesis:_(_(_(f_._p2)_`2_>=_0_&_LE_f_._p2,f_._p3,P_)_or_(_(f_._p2)_`2_<_0_&_LE_f_._p2,f_._p3,P_)_) percases ( (f . p2) `2 >= 0 or (f . p2) `2 < 0 ) ; case (f . p2) `2 >= 0 ; ::_thesis: LE f . p2,f . p3,P then f . p2 in { p9 where p9 is Point of (TOP-REAL 2) : ( p9 in P & p9 `2 >= 0 ) } by A13; then f . p2 in Upper_Arc P by A8, JGRAPH_5:34; hence LE f . p2,f . p3,P by A77, A78, JORDAN6:def_10; ::_thesis: verum end; caseA79: (f . p2) `2 < 0 ; ::_thesis: LE f . p2,f . p3,P ((p2 `1) * (sqrt (1 + ((p3 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) >= ((p3 `1) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by A51, A52, A53, A68, A69, A71, SQUARE_1:56, XREAL_1:72; then p2 `1 >= ((p3 `1) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by A71, XCMPLX_1:89; then (p2 `1) / (sqrt (1 + ((p2 `2) ^2))) >= (((p3 `1) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p3 `1) ^2)))) / (sqrt (1 + ((p2 `2) ^2))) by A54, XREAL_1:72; then (p2 `1) / (sqrt (1 + ((p2 `2) ^2))) >= (((p3 `1) * (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p2 `2) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by XCMPLX_1:48; then (p2 `1) / (sqrt (1 + ((p2 `2) ^2))) >= (p3 `1) / (sqrt (1 + ((p3 `1) ^2))) by A54, XCMPLX_1:89; hence LE f . p2,f . p3,P by A8, A13, A14, A56, A67, A71, A74, A75, A78, A79, JGRAPH_5:56; ::_thesis: verum end; end; end; hence LE f . p2,f . p3,P ; ::_thesis: verum end; end; end; hence LE f . p2,f . p3,P ; ::_thesis: verum end; caseA80: ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> |[(- 1),(- 1)]| ) ; ::_thesis: LE f . p2,f . p3,P then A81: p2 `2 = - 1 by Th3; A82: - 1 <= p2 `1 by A80, Th3; (p2 `1) ^2 >= 0 by XREAL_1:63; then A83: sqrt (1 + ((p2 `1) ^2)) > 0 by SQUARE_1:25; A84: - (p2 `2) >= p2 `1 by A80, A81, Th3; p2 <> 0. (TOP-REAL 2) by A81, EUCLID:52, EUCLID:54; then A85: f . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| by A2, A81, A82, A84, JGRAPH_3:4; then A86: (f . p2) `1 = (p2 `1) / (sqrt (1 + (((p2 `1) / (- 1)) ^2))) by A81, EUCLID:52 .= (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) ; A87: (f . p2) `2 = (p2 `2) / (sqrt (1 + (((p2 `1) / (- 1)) ^2))) by A81, A85, EUCLID:52 .= (p2 `2) / (sqrt (1 + ((p2 `1) ^2))) ; A88: W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; then A89: p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) by A6, A80, Th62; A90: p2 `1 >= p3 `1 by A6, A80, A88, Th62; A91: p3 `2 = - 1 by A89, Th3; A92: - 1 <= p3 `1 by A89, Th3; A93: (p3 `1) ^2 >= 0 by XREAL_1:63; then A94: sqrt (1 + ((p3 `1) ^2)) > 0 by SQUARE_1:25; A95: - (p3 `2) >= p3 `1 by A89, A91, Th3; p3 <> 0. (TOP-REAL 2) by A91, EUCLID:52, EUCLID:54; then A96: f . p3 = |[((p3 `1) / (sqrt (1 + (((p3 `1) / (p3 `2)) ^2)))),((p3 `2) / (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))))]| by A2, A91, A92, A95, JGRAPH_3:4; then A97: (f . p3) `1 = (p3 `1) / (sqrt (1 + (((p3 `1) / (- 1)) ^2))) by A91, EUCLID:52 .= (p3 `1) / (sqrt (1 + ((p3 `1) ^2))) ; (f . p3) `2 = (p3 `2) / (sqrt (1 + (((p3 `1) / (- 1)) ^2))) by A91, A96, EUCLID:52 .= (p3 `2) / (sqrt (1 + ((p3 `1) ^2))) ; then A98: (f . p3) `2 < 0 by A91, A93, SQUARE_1:25, XREAL_1:141; W-min P = |[(- 1),0]| by A8, JGRAPH_5:29; then A99: f . p3 <> W-min P by A98, EUCLID:52; (p2 `1) * (sqrt (1 + ((p3 `1) ^2))) >= (p3 `1) * (sqrt (1 + ((p2 `1) ^2))) by A90, SQUARE_1:57; then ((p2 `1) * (sqrt (1 + ((p3 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) >= ((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by A94, XREAL_1:72; then p2 `1 >= ((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by A94, XCMPLX_1:89; then (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) >= (((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2)))) / (sqrt (1 + ((p2 `1) ^2))) by A83, XREAL_1:72; then (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) >= (((p3 `1) * (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p2 `1) ^2)))) / (sqrt (1 + ((p3 `1) ^2))) by XCMPLX_1:48; then (p2 `1) / (sqrt (1 + ((p2 `1) ^2))) >= (p3 `1) / (sqrt (1 + ((p3 `1) ^2))) by A83, XCMPLX_1:89; hence LE f . p2,f . p3,P by A8, A13, A14, A81, A83, A86, A87, A97, A98, A99, JGRAPH_5:56; ::_thesis: verum end; end; end; hence LE f . p2,f . p3,P ; ::_thesis: verum end; theorem Th67: :: JGRAPH_6:67 for p being Point of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ & p `1 = - 1 & p `2 < 0 holds ( (f . p) `1 < 0 & (f . p) `2 < 0 ) proof let p be Point of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ & p `1 = - 1 & p `2 < 0 holds ( (f . p) `1 < 0 & (f . p) `2 < 0 ) let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = Sq_Circ & p `1 = - 1 & p `2 < 0 implies ( (f . p) `1 < 0 & (f . p) `2 < 0 ) ) assume that A1: f = Sq_Circ and A2: p `1 = - 1 and A3: p `2 < 0 ; ::_thesis: ( (f . p) `1 < 0 & (f . p) `2 < 0 ) now__::_thesis:_(_(_p_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_p_<>_0._(TOP-REAL_2)_&_(f_._p)_`1_<_0_&_(f_._p)_`2_<_0_)_) percases ( p = 0. (TOP-REAL 2) or p <> 0. (TOP-REAL 2) ) ; case p = 0. (TOP-REAL 2) ; ::_thesis: contradiction hence contradiction by A2, EUCLID:52, EUCLID:54; ::_thesis: verum end; caseA4: p <> 0. (TOP-REAL 2) ; ::_thesis: ( (f . p) `1 < 0 & (f . p) `2 < 0 ) now__::_thesis:_(_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_&_(f_._p)_`1_<_0_&_(f_._p)_`2_<_0_)_or_(_not_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_&_not_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_&_(f_._p)_`1_<_0_&_(f_._p)_`2_<_0_)_) percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ) ; case ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: ( (f . p) `1 < 0 & (f . p) `2 < 0 ) then A5: f . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A1, A4, JGRAPH_3:def_1; then A6: (f . p) `1 = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52; (f . p) `2 = (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) by A5, EUCLID:52; hence ( (f . p) `1 < 0 & (f . p) `2 < 0 ) by A2, A3, A6, SQUARE_1:25, XREAL_1:141; ::_thesis: verum end; case ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: ( (f . p) `1 < 0 & (f . p) `2 < 0 ) then A7: f . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A1, A4, JGRAPH_3:def_1; then A8: (f . p) `1 = (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52; (f . p) `2 = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) by A7, EUCLID:52; hence ( (f . p) `1 < 0 & (f . p) `2 < 0 ) by A2, A3, A8, SQUARE_1:25, XREAL_1:141; ::_thesis: verum end; end; end; hence ( (f . p) `1 < 0 & (f . p) `2 < 0 ) ; ::_thesis: verum end; end; end; hence ( (f . p) `1 < 0 & (f . p) `2 < 0 ) ; ::_thesis: verum end; theorem Th68: :: JGRAPH_6:68 for p being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds ( (f . p) `1 >= 0 iff p `1 >= 0 ) proof let p be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds ( (f . p) `1 >= 0 iff p `1 >= 0 ) let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds ( (f . p) `1 >= 0 iff p `1 >= 0 ) let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = Sq_Circ implies ( (f . p) `1 >= 0 iff p `1 >= 0 ) ) assume A1: f = Sq_Circ ; ::_thesis: ( (f . p) `1 >= 0 iff p `1 >= 0 ) thus ( (f . p) `1 >= 0 implies p `1 >= 0 ) ::_thesis: ( p `1 >= 0 implies (f . p) `1 >= 0 ) proof assume A2: (f . p) `1 >= 0 ; ::_thesis: p `1 >= 0 reconsider g = Sq_Circ " as Function of (TOP-REAL 2),(TOP-REAL 2) by JGRAPH_3:29; A3: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; set q = f . p; now__::_thesis:_(_(_f_._p_=_0._(TOP-REAL_2)_&_(g_._(f_._p))_`1_>=_0_)_or_(_f_._p_<>_0._(TOP-REAL_2)_&_(g_._(f_._p))_`1_>=_0_)_) percases ( f . p = 0. (TOP-REAL 2) or f . p <> 0. (TOP-REAL 2) ) ; case f . p = 0. (TOP-REAL 2) ; ::_thesis: (g . (f . p)) `1 >= 0 hence (g . (f . p)) `1 >= 0 by A2, JGRAPH_3:28; ::_thesis: verum end; caseA4: f . p <> 0. (TOP-REAL 2) ; ::_thesis: (g . (f . p)) `1 >= 0 now__::_thesis:_(_(_(_(_(f_._p)_`2_<=_(f_._p)_`1_&_-_((f_._p)_`1)_<=_(f_._p)_`2_)_or_(_(f_._p)_`2_>=_(f_._p)_`1_&_(f_._p)_`2_<=_-_((f_._p)_`1)_)_)_&_(g_._(f_._p))_`1_>=_0_)_or_(_not_(_(f_._p)_`2_<=_(f_._p)_`1_&_-_((f_._p)_`1)_<=_(f_._p)_`2_)_&_not_(_(f_._p)_`2_>=_(f_._p)_`1_&_(f_._p)_`2_<=_-_((f_._p)_`1)_)_&_(g_._(f_._p))_`1_>=_0_)_) percases ( ( (f . p) `2 <= (f . p) `1 & - ((f . p) `1) <= (f . p) `2 ) or ( (f . p) `2 >= (f . p) `1 & (f . p) `2 <= - ((f . p) `1) ) or ( not ( (f . p) `2 <= (f . p) `1 & - ((f . p) `1) <= (f . p) `2 ) & not ( (f . p) `2 >= (f . p) `1 & (f . p) `2 <= - ((f . p) `1) ) ) ) ; case ( ( (f . p) `2 <= (f . p) `1 & - ((f . p) `1) <= (f . p) `2 ) or ( (f . p) `2 >= (f . p) `1 & (f . p) `2 <= - ((f . p) `1) ) ) ; ::_thesis: (g . (f . p)) `1 >= 0 then A5: g . (f . p) = |[(((f . p) `1) * (sqrt (1 + ((((f . p) `2) / ((f . p) `1)) ^2)))),(((f . p) `2) * (sqrt (1 + ((((f . p) `2) / ((f . p) `1)) ^2))))]| by A4, JGRAPH_3:28; (((f . p) `2) / ((f . p) `1)) ^2 >= 0 by XREAL_1:63; then sqrt (1 + ((((f . p) `2) / ((f . p) `1)) ^2)) > 0 by SQUARE_1:25; hence (g . (f . p)) `1 >= 0 by A2, A5, EUCLID:52; ::_thesis: verum end; case ( not ( (f . p) `2 <= (f . p) `1 & - ((f . p) `1) <= (f . p) `2 ) & not ( (f . p) `2 >= (f . p) `1 & (f . p) `2 <= - ((f . p) `1) ) ) ; ::_thesis: (g . (f . p)) `1 >= 0 then A6: g . (f . p) = |[(((f . p) `1) * (sqrt (1 + ((((f . p) `1) / ((f . p) `2)) ^2)))),(((f . p) `2) * (sqrt (1 + ((((f . p) `1) / ((f . p) `2)) ^2))))]| by JGRAPH_3:28; (((f . p) `1) / ((f . p) `2)) ^2 >= 0 by XREAL_1:63; then sqrt (1 + ((((f . p) `1) / ((f . p) `2)) ^2)) > 0 by SQUARE_1:25; hence (g . (f . p)) `1 >= 0 by A2, A6, EUCLID:52; ::_thesis: verum end; end; end; hence (g . (f . p)) `1 >= 0 ; ::_thesis: verum end; end; end; hence p `1 >= 0 by A1, A3, FUNCT_1:34; ::_thesis: verum end; thus ( p `1 >= 0 implies (f . p) `1 >= 0 ) ::_thesis: verum proof assume A7: p `1 >= 0 ; ::_thesis: (f . p) `1 >= 0 now__::_thesis:_(_(_p_=_0._(TOP-REAL_2)_&_(f_._p)_`1_>=_0_)_or_(_p_<>_0._(TOP-REAL_2)_&_(f_._p)_`1_>=_0_)_) percases ( p = 0. (TOP-REAL 2) or p <> 0. (TOP-REAL 2) ) ; case p = 0. (TOP-REAL 2) ; ::_thesis: (f . p) `1 >= 0 hence (f . p) `1 >= 0 by A1, A7, JGRAPH_3:def_1; ::_thesis: verum end; caseA8: p <> 0. (TOP-REAL 2) ; ::_thesis: (f . p) `1 >= 0 now__::_thesis:_(_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_&_(f_._p)_`1_>=_0_)_or_(_not_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_&_not_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_&_(f_._p)_`1_>=_0_)_) percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ) ; case ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: (f . p) `1 >= 0 then A9: f . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A1, A8, JGRAPH_3:def_1; ((p `2) / (p `1)) ^2 >= 0 by XREAL_1:63; then sqrt (1 + (((p `2) / (p `1)) ^2)) > 0 by SQUARE_1:25; hence (f . p) `1 >= 0 by A7, A9, EUCLID:52; ::_thesis: verum end; case ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: (f . p) `1 >= 0 then A10: f . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A1, A8, JGRAPH_3:def_1; ((p `1) / (p `2)) ^2 >= 0 by XREAL_1:63; then sqrt (1 + (((p `1) / (p `2)) ^2)) > 0 by SQUARE_1:25; hence (f . p) `1 >= 0 by A7, A10, EUCLID:52; ::_thesis: verum end; end; end; hence (f . p) `1 >= 0 ; ::_thesis: verum end; end; end; hence (f . p) `1 >= 0 ; ::_thesis: verum end; end; theorem Th69: :: JGRAPH_6:69 for p being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds ( (f . p) `2 >= 0 iff p `2 >= 0 ) proof let p be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds ( (f . p) `2 >= 0 iff p `2 >= 0 ) let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds ( (f . p) `2 >= 0 iff p `2 >= 0 ) let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = Sq_Circ implies ( (f . p) `2 >= 0 iff p `2 >= 0 ) ) assume A1: f = Sq_Circ ; ::_thesis: ( (f . p) `2 >= 0 iff p `2 >= 0 ) thus ( (f . p) `2 >= 0 implies p `2 >= 0 ) ::_thesis: ( p `2 >= 0 implies (f . p) `2 >= 0 ) proof assume A2: (f . p) `2 >= 0 ; ::_thesis: p `2 >= 0 reconsider g = Sq_Circ " as Function of (TOP-REAL 2),(TOP-REAL 2) by JGRAPH_3:29; A3: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; set q = f . p; now__::_thesis:_(_(_f_._p_=_0._(TOP-REAL_2)_&_(g_._(f_._p))_`2_>=_0_)_or_(_f_._p_<>_0._(TOP-REAL_2)_&_(g_._(f_._p))_`2_>=_0_)_) percases ( f . p = 0. (TOP-REAL 2) or f . p <> 0. (TOP-REAL 2) ) ; case f . p = 0. (TOP-REAL 2) ; ::_thesis: (g . (f . p)) `2 >= 0 hence (g . (f . p)) `2 >= 0 by A2, JGRAPH_3:28; ::_thesis: verum end; caseA4: f . p <> 0. (TOP-REAL 2) ; ::_thesis: (g . (f . p)) `2 >= 0 now__::_thesis:_(_(_(_(_(f_._p)_`2_<=_(f_._p)_`1_&_-_((f_._p)_`1)_<=_(f_._p)_`2_)_or_(_(f_._p)_`2_>=_(f_._p)_`1_&_(f_._p)_`2_<=_-_((f_._p)_`1)_)_)_&_(g_._(f_._p))_`2_>=_0_)_or_(_not_(_(f_._p)_`2_<=_(f_._p)_`1_&_-_((f_._p)_`1)_<=_(f_._p)_`2_)_&_not_(_(f_._p)_`2_>=_(f_._p)_`1_&_(f_._p)_`2_<=_-_((f_._p)_`1)_)_&_(g_._(f_._p))_`2_>=_0_)_) percases ( ( (f . p) `2 <= (f . p) `1 & - ((f . p) `1) <= (f . p) `2 ) or ( (f . p) `2 >= (f . p) `1 & (f . p) `2 <= - ((f . p) `1) ) or ( not ( (f . p) `2 <= (f . p) `1 & - ((f . p) `1) <= (f . p) `2 ) & not ( (f . p) `2 >= (f . p) `1 & (f . p) `2 <= - ((f . p) `1) ) ) ) ; case ( ( (f . p) `2 <= (f . p) `1 & - ((f . p) `1) <= (f . p) `2 ) or ( (f . p) `2 >= (f . p) `1 & (f . p) `2 <= - ((f . p) `1) ) ) ; ::_thesis: (g . (f . p)) `2 >= 0 then A5: g . (f . p) = |[(((f . p) `1) * (sqrt (1 + ((((f . p) `2) / ((f . p) `1)) ^2)))),(((f . p) `2) * (sqrt (1 + ((((f . p) `2) / ((f . p) `1)) ^2))))]| by A4, JGRAPH_3:28; (((f . p) `2) / ((f . p) `1)) ^2 >= 0 by XREAL_1:63; then sqrt (1 + ((((f . p) `2) / ((f . p) `1)) ^2)) > 0 by SQUARE_1:25; hence (g . (f . p)) `2 >= 0 by A2, A5, EUCLID:52; ::_thesis: verum end; case ( not ( (f . p) `2 <= (f . p) `1 & - ((f . p) `1) <= (f . p) `2 ) & not ( (f . p) `2 >= (f . p) `1 & (f . p) `2 <= - ((f . p) `1) ) ) ; ::_thesis: (g . (f . p)) `2 >= 0 then A6: g . (f . p) = |[(((f . p) `1) * (sqrt (1 + ((((f . p) `1) / ((f . p) `2)) ^2)))),(((f . p) `2) * (sqrt (1 + ((((f . p) `1) / ((f . p) `2)) ^2))))]| by JGRAPH_3:28; (((f . p) `1) / ((f . p) `2)) ^2 >= 0 by XREAL_1:63; then sqrt (1 + ((((f . p) `1) / ((f . p) `2)) ^2)) > 0 by SQUARE_1:25; hence (g . (f . p)) `2 >= 0 by A2, A6, EUCLID:52; ::_thesis: verum end; end; end; hence (g . (f . p)) `2 >= 0 ; ::_thesis: verum end; end; end; hence p `2 >= 0 by A1, A3, FUNCT_1:34; ::_thesis: verum end; thus ( p `2 >= 0 implies (f . p) `2 >= 0 ) ::_thesis: verum proof assume A7: p `2 >= 0 ; ::_thesis: (f . p) `2 >= 0 now__::_thesis:_(_(_p_=_0._(TOP-REAL_2)_&_(f_._p)_`2_>=_0_)_or_(_p_<>_0._(TOP-REAL_2)_&_(f_._p)_`2_>=_0_)_) percases ( p = 0. (TOP-REAL 2) or p <> 0. (TOP-REAL 2) ) ; case p = 0. (TOP-REAL 2) ; ::_thesis: (f . p) `2 >= 0 hence (f . p) `2 >= 0 by A1, A7, JGRAPH_3:def_1; ::_thesis: verum end; caseA8: p <> 0. (TOP-REAL 2) ; ::_thesis: (f . p) `2 >= 0 now__::_thesis:_(_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_&_(f_._p)_`2_>=_0_)_or_(_not_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_&_not_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_&_(f_._p)_`2_>=_0_)_) percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ) ; case ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: (f . p) `2 >= 0 then A9: f . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A1, A8, JGRAPH_3:def_1; ((p `2) / (p `1)) ^2 >= 0 by XREAL_1:63; then sqrt (1 + (((p `2) / (p `1)) ^2)) > 0 by SQUARE_1:25; hence (f . p) `2 >= 0 by A7, A9, EUCLID:52; ::_thesis: verum end; case ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: (f . p) `2 >= 0 then A10: f . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A1, A8, JGRAPH_3:def_1; ((p `1) / (p `2)) ^2 >= 0 by XREAL_1:63; then sqrt (1 + (((p `1) / (p `2)) ^2)) > 0 by SQUARE_1:25; hence (f . p) `2 >= 0 by A7, A10, EUCLID:52; ::_thesis: verum end; end; end; hence (f . p) `2 >= 0 ; ::_thesis: verum end; end; end; hence (f . p) `2 >= 0 ; ::_thesis: verum end; end; theorem Th70: :: JGRAPH_6:70 for p, q being Point of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ & p in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & q in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) holds (f . p) `1 <= (f . q) `1 proof let p, q be Point of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ & p in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & q in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) holds (f . p) `1 <= (f . q) `1 let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = Sq_Circ & p in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & q in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) implies (f . p) `1 <= (f . q) `1 ) assume that A1: f = Sq_Circ and A2: p in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) and A3: q in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ; ::_thesis: (f . p) `1 <= (f . q) `1 A4: p `1 = - 1 by A2, Th1; A5: - 1 <= p `2 by A2, Th1; A6: p `2 <= 1 by A2, Th1; A7: q `2 = - 1 by A3, Th3; A8: - 1 <= q `1 by A3, Th3; A9: q `1 <= 1 by A3, Th3; A10: p <> 0. (TOP-REAL 2) by A4, EUCLID:52, EUCLID:54; A11: q <> 0. (TOP-REAL 2) by A7, EUCLID:52, EUCLID:54; p `2 <= - (p `1) by A2, A4, Th1; then f . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A1, A4, A5, A10, JGRAPH_3:def_1; then A12: (f . p) `1 = (- 1) / (sqrt (1 + (((p `2) / (- 1)) ^2))) by A4, EUCLID:52 .= (- 1) / (sqrt (1 + ((p `2) ^2))) ; (p `2) ^2 >= 0 by XREAL_1:63; then A13: sqrt (1 + ((p `2) ^2)) > 0 by SQUARE_1:25; (q `1) ^2 >= 0 by XREAL_1:63; then A14: sqrt (1 + ((q `1) ^2)) > 0 by SQUARE_1:25; q `1 <= - (q `2) by A3, A7, Th3; then f . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by A1, A7, A8, A11, JGRAPH_3:4; then A15: (f . q) `1 = (q `1) / (sqrt (1 + (((q `1) / (- 1)) ^2))) by A7, EUCLID:52 .= (q `1) / (sqrt (1 + ((q `1) ^2))) ; - (sqrt (1 + ((q `1) ^2))) <= (q `1) * (sqrt (1 + ((p `2) ^2))) by A5, A6, A8, A9, SQUARE_1:55; then ((- 1) * (sqrt (1 + ((q `1) ^2)))) / (sqrt (1 + ((q `1) ^2))) <= ((q `1) * (sqrt (1 + ((p `2) ^2)))) / (sqrt (1 + ((q `1) ^2))) by A14, XREAL_1:72; then - 1 <= ((q `1) * (sqrt (1 + ((p `2) ^2)))) / (sqrt (1 + ((q `1) ^2))) by A14, XCMPLX_1:89; then - 1 <= ((q `1) / (sqrt (1 + ((q `1) ^2)))) * (sqrt (1 + ((p `2) ^2))) by XCMPLX_1:74; then (- 1) / (sqrt (1 + ((p `2) ^2))) <= (((q `1) / (sqrt (1 + ((q `1) ^2)))) * (sqrt (1 + ((p `2) ^2)))) / (sqrt (1 + ((p `2) ^2))) by A13, XREAL_1:72; hence (f . p) `1 <= (f . q) `1 by A12, A13, A15, XCMPLX_1:89; ::_thesis: verum end; theorem Th71: :: JGRAPH_6:71 for p, q being Point of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ & p in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & q in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p `2 >= q `2 & p `2 < 0 holds (f . p) `2 >= (f . q) `2 proof let p, q be Point of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ & p in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & q in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p `2 >= q `2 & p `2 < 0 holds (f . p) `2 >= (f . q) `2 let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = Sq_Circ & p in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & q in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & p `2 >= q `2 & p `2 < 0 implies (f . p) `2 >= (f . q) `2 ) assume that A1: f = Sq_Circ and A2: p in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) and A3: q in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) and A4: p `2 >= q `2 and A5: p `2 < 0 ; ::_thesis: (f . p) `2 >= (f . q) `2 A6: p `1 = - 1 by A2, Th1; A7: - 1 <= p `2 by A2, Th1; (p `2) ^2 >= 0 by XREAL_1:63; then A8: sqrt (1 + ((p `2) ^2)) > 0 by SQUARE_1:25; (q `2) ^2 >= 0 by XREAL_1:63; then A9: sqrt (1 + ((q `2) ^2)) > 0 by SQUARE_1:25; A10: p `2 <= - (p `1) by A5, A6; p <> 0. (TOP-REAL 2) by A5, EUCLID:52, EUCLID:54; then f . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A1, A6, A7, A10, JGRAPH_3:def_1; then A11: (f . p) `2 = (p `2) / (sqrt (1 + (((p `2) / (- 1)) ^2))) by A6, EUCLID:52 .= (p `2) / (sqrt (1 + ((p `2) ^2))) ; A12: q `1 = - 1 by A3, Th1; A13: - 1 <= q `2 by A3, Th1; A14: q `2 <= - (q `1) by A4, A5, A12; q <> 0. (TOP-REAL 2) by A4, A5, EUCLID:52, EUCLID:54; then f . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by A1, A12, A13, A14, JGRAPH_3:def_1; then A15: (f . q) `2 = (q `2) / (sqrt (1 + (((q `2) / (- 1)) ^2))) by A12, EUCLID:52 .= (q `2) / (sqrt (1 + ((q `2) ^2))) ; (p `2) * (sqrt (1 + ((q `2) ^2))) >= (q `2) * (sqrt (1 + ((p `2) ^2))) by A4, A5, Lm2; then ((p `2) * (sqrt (1 + ((q `2) ^2)))) / (sqrt (1 + ((q `2) ^2))) >= ((q `2) * (sqrt (1 + ((p `2) ^2)))) / (sqrt (1 + ((q `2) ^2))) by A9, XREAL_1:72; then p `2 >= ((q `2) * (sqrt (1 + ((p `2) ^2)))) / (sqrt (1 + ((q `2) ^2))) by A9, XCMPLX_1:89; then (p `2) / (sqrt (1 + ((p `2) ^2))) >= (((q `2) * (sqrt (1 + ((p `2) ^2)))) / (sqrt (1 + ((q `2) ^2)))) / (sqrt (1 + ((p `2) ^2))) by A8, XREAL_1:72; then (p `2) / (sqrt (1 + ((p `2) ^2))) >= (((q `2) * (sqrt (1 + ((p `2) ^2)))) / (sqrt (1 + ((p `2) ^2)))) / (sqrt (1 + ((q `2) ^2))) by XCMPLX_1:48; hence (f . p) `2 >= (f . q) `2 by A8, A11, A15, XCMPLX_1:89; ::_thesis: verum end; theorem Th72: :: JGRAPH_6:72 for p1, p2, p3, p4 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) holds f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P proof let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) holds f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) holds f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( P = circle (0,0,1) & f = Sq_Circ & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) implies f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) set K = rectangle ((- 1),1,(- 1),1); assume that A1: P = circle (0,0,1) and A2: f = Sq_Circ ; ::_thesis: ( not LE p1,p2, rectangle ((- 1),1,(- 1),1) or not LE p2,p3, rectangle ((- 1),1,(- 1),1) or not LE p3,p4, rectangle ((- 1),1,(- 1),1) or f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) A3: rectangle ((- 1),1,(- 1),1) is being_simple_closed_curve by Th50; A4: rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } by Lm15; A5: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } by A1, Th24; thus ( LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) implies f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) ::_thesis: verum proof assume that A6: LE p1,p2, rectangle ((- 1),1,(- 1),1) and A7: LE p2,p3, rectangle ((- 1),1,(- 1),1) and A8: LE p3,p4, rectangle ((- 1),1,(- 1),1) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P A9: p1 in rectangle ((- 1),1,(- 1),1) by A3, A6, JORDAN7:5; A10: p2 in rectangle ((- 1),1,(- 1),1) by A3, A6, JORDAN7:5; A11: p3 in rectangle ((- 1),1,(- 1),1) by A3, A7, JORDAN7:5; A12: p4 in rectangle ((- 1),1,(- 1),1) by A3, A8, JORDAN7:5; then A13: ex q8 being Point of (TOP-REAL 2) st ( q8 = p4 & ( ( q8 `1 = - 1 & - 1 <= q8 `2 & q8 `2 <= 1 ) or ( q8 `2 = 1 & - 1 <= q8 `1 & q8 `1 <= 1 ) or ( q8 `1 = 1 & - 1 <= q8 `2 & q8 `2 <= 1 ) or ( q8 `2 = - 1 & - 1 <= q8 `1 & q8 `1 <= 1 ) ) ) by A4; A14: LE p1,p3, rectangle ((- 1),1,(- 1),1) by A6, A7, Th50, JORDAN6:58; A15: LE p2,p4, rectangle ((- 1),1,(- 1),1) by A7, A8, Th50, JORDAN6:58; A16: W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; A17: |[(- 1),0]| `2 = 0 by EUCLID:52; A18: (1 / 2) * (|[(- 1),(- 1)]| + |[(- 1),1]|) = ((1 / 2) * |[(- 1),(- 1)]|) + ((1 / 2) * |[(- 1),1]|) by EUCLID:32 .= |[((1 / 2) * (- 1)),((1 / 2) * (- 1))]| + ((1 / 2) * |[(- 1),1]|) by EUCLID:58 .= |[((1 / 2) * (- 1)),((1 / 2) * (- 1))]| + |[((1 / 2) * (- 1)),((1 / 2) * 1)]| by EUCLID:58 .= |[(((1 / 2) * (- 1)) + ((1 / 2) * (- 1))),(((1 / 2) * (- 1)) + ((1 / 2) * 1))]| by EUCLID:56 .= |[(- 1),0]| ; then A19: |[(- 1),0]| in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) by RLTOPSP1:69; now__::_thesis:_(_(_p1_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_p1_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_p1_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_p1_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p1_<>_W-min_(rectangle_((-_1),1,(-_1),1))_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_) percases ( p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) or p1 in LSeg (|[(- 1),1]|,|[1,1]|) or p1 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p1 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p1 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A9, A16, Th63, RLTOPSP1:68; caseA20: p1 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P then A21: p1 `1 = - 1 by Th1; then A22: (f . p1) `1 < 0 by A2, Th68; A23: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A24: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A25: f . p1 in P by A9, A23, FUNCT_1:def_6; now__::_thesis:_(_(_p1_`2_>=_0_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_p1_`2_<_0_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_) percases ( p1 `2 >= 0 or p1 `2 < 0 ) ; caseA26: p1 `2 >= 0 ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P then A27: LE f . p1,f . p2,P by A1, A2, A6, A20, Th65; A28: LE f . p2,f . p3,P by A1, A2, A6, A7, A20, A26, Th66; LE f . p3,f . p4,P by A1, A2, A8, A14, A20, A26, Th66; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A27, A28, JORDAN17:def_1; ::_thesis: verum end; caseA29: p1 `2 < 0 ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P now__::_thesis:_(_(_p2_`2_<_0_&_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_(_not_p2_`2_<_0_or_not_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_)_&_rectangle_((-_1),1,(-_1),1)_=__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_(_p_`1_=_-_1_&_-_1_<=_p_`2_&_p_`2_<=_1_)_or_(_p_`1_=_1_&_-_1_<=_p_`2_&_p_`2_<=_1_)_or_(_-_1_=_p_`2_&_-_1_<=_p_`1_&_p_`1_<=_1_)_or_(_1_=_p_`2_&_-_1_<=_p_`1_&_p_`1_<=_1_)_)__}__&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_) percases ( ( p2 `2 < 0 & p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) or not p2 `2 < 0 or not p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; caseA30: ( p2 `2 < 0 & p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P then A31: p2 `1 = - 1 by Th1; A32: f . p2 in P by A10, A23, A24, FUNCT_1:def_6; A33: p1 `2 <= p2 `2 by A6, A20, A30, Th55; now__::_thesis:_(_(_p3_`2_<_0_&_p3_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_(_not_p3_`2_<_0_or_not_p3_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_)_&_rectangle_((-_1),1,(-_1),1)_=__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_(_p_`1_=_-_1_&_-_1_<=_p_`2_&_p_`2_<=_1_)_or_(_p_`1_=_1_&_-_1_<=_p_`2_&_p_`2_<=_1_)_or_(_-_1_=_p_`2_&_-_1_<=_p_`1_&_p_`1_<=_1_)_or_(_1_=_p_`2_&_-_1_<=_p_`1_&_p_`1_<=_1_)_)__}__&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_) percases ( ( p3 `2 < 0 & p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) or not p3 `2 < 0 or not p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; caseA34: ( p3 `2 < 0 & p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P then A35: p3 `1 = - 1 by Th1; A36: f . p3 in P by A11, A23, A24, FUNCT_1:def_6; A37: p2 `2 <= p3 `2 by A7, A30, A34, Th55; now__::_thesis:_(_(_p4_`2_<_0_&_p4_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_(_not_p4_`2_<_0_or_not_p4_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_)_&_rectangle_((-_1),1,(-_1),1)_=__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_(_p_`1_=_-_1_&_-_1_<=_p_`2_&_p_`2_<=_1_)_or_(_p_`1_=_1_&_-_1_<=_p_`2_&_p_`2_<=_1_)_or_(_-_1_=_p_`2_&_-_1_<=_p_`1_&_p_`1_<=_1_)_or_(_1_=_p_`2_&_-_1_<=_p_`1_&_p_`1_<=_1_)_)__}__&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_) percases ( ( p4 `2 < 0 & p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) or not p4 `2 < 0 or not p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; caseA38: ( p4 `2 < 0 & p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P then A39: p4 `1 = - 1 by Th1; A40: (f . p2) `1 < 0 by A2, A30, A31, Th67; A41: (f . p2) `2 < 0 by A2, A30, A31, Th67; A42: (f . p3) `1 < 0 by A2, A34, A35, Th67; A43: (f . p3) `2 < 0 by A2, A34, A35, Th67; A44: (f . p4) `1 < 0 by A2, A38, A39, Th67; A45: (f . p4) `2 < 0 by A2, A38, A39, Th67; (f . p1) `2 <= (f . p2) `2 by A2, A20, A30, A33, Th71; then A46: LE f . p1,f . p2,P by A5, A22, A25, A32, A40, A41, JGRAPH_5:51; (f . p2) `2 <= (f . p3) `2 by A2, A30, A34, A37, Th71; then A47: LE f . p2,f . p3,P by A5, A32, A36, A40, A42, A43, JGRAPH_5:51; A48: f . p4 in P by A12, A23, A24, FUNCT_1:def_6; p3 `2 <= p4 `2 by A8, A34, A38, Th55; then (f . p3) `2 <= (f . p4) `2 by A2, A34, A38, Th71; then LE f . p3,f . p4,P by A5, A36, A42, A44, A45, A48, JGRAPH_5:51; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A46, A47, JORDAN17:def_1; ::_thesis: verum end; caseA49: ( not p4 `2 < 0 or not p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; ::_thesis: ( rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } & f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) A50: now__::_thesis:_(_(_p4_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_(_(_p4_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p4_`2_)_or_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p4_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_(_(_p4_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p4_`2_)_or_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p4_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_(_(_p4_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p4_`2_)_or_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p4_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_(_(_p4_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p4_`2_)_or_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p4_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_) percases ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A12, Th63; case p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ; ::_thesis: ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A49, EUCLID:52; ::_thesis: verum end; case p4 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) ; ::_thesis: verum end; case p4 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) ; ::_thesis: verum end; caseA51: p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ; ::_thesis: ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) A52: W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; now__::_thesis:_not_p4_=_W-min_(rectangle_((-_1),1,(-_1),1)) assume A53: p4 = W-min (rectangle ((- 1),1,(- 1),1)) ; ::_thesis: contradiction then p4 `2 = - 1 by A52, EUCLID:52; hence contradiction by A49, A52, A53, RLTOPSP1:68; ::_thesis: verum end; hence ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A51; ::_thesis: verum end; end; end; A54: (f . p2) `1 < 0 by A2, A30, A31, Th67; A55: (f . p2) `2 < 0 by A2, A30, A31, Th67; A56: (f . p3) `1 < 0 by A2, A34, A35, Th67; A57: (f . p3) `2 < 0 by A2, A34, A35, Th67; (f . p1) `2 <= (f . p2) `2 by A2, A20, A30, A33, Th71; then A58: LE f . p1,f . p2,P by A5, A22, A25, A32, A54, A55, JGRAPH_5:51; (f . p2) `2 <= (f . p3) `2 by A2, A30, A34, A37, Th71; then A59: LE f . p2,f . p3,P by A5, A32, A36, A54, A56, A57, JGRAPH_5:51; A60: now__::_thesis:_(_(_p4_`1_=_-_1_&_p4_`2_<_0_&_p1_`2_<=_p4_`2_&_contradiction_)_or_(_(_not_p4_`1_=_-_1_or_not_p4_`2_<_0_or_not_p1_`2_<=_p4_`2_)_&_LE_f_._p4,f_._p1,P_)_) percases ( ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 <= p4 `2 ) or not p4 `1 = - 1 or not p4 `2 < 0 or not p1 `2 <= p4 `2 ) ; caseA61: ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 <= p4 `2 ) ; ::_thesis: contradiction now__::_thesis:_(_(_p4_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p4_`2_&_contradiction_)_or_(_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_contradiction_)_or_(_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_contradiction_)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p4_<>_W-min_(rectangle_((-_1),1,(-_1),1))_&_contradiction_)_) percases ( ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A50; case ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p4 `2 ) ; ::_thesis: contradiction hence contradiction by A61, EUCLID:52; ::_thesis: verum end; case p4 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: contradiction hence contradiction by A61, Th3; ::_thesis: verum end; case p4 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: contradiction hence contradiction by A61, Th1; ::_thesis: verum end; caseA62: ( p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p4 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ; ::_thesis: contradiction then A63: p4 `2 = - 1 by Th3; A64: W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; then A65: (W-min (rectangle ((- 1),1,(- 1),1))) `1 = - 1 by EUCLID:52; (W-min (rectangle ((- 1),1,(- 1),1))) `2 = - 1 by A64, EUCLID:52; hence contradiction by A61, A62, A63, A65, TOPREAL3:6; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA66: ( not p4 `1 = - 1 or not p4 `2 < 0 or not p1 `2 <= p4 `2 ) ; ::_thesis: LE f . p4,f . p1,P A67: ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A12, Th63; now__::_thesis:_(_(_p4_`1_<>_-_1_&_LE_f_._p4,f_._p1,P_)_or_(_p4_`1_=_-_1_&_p4_`2_>=_0_&_LE_f_._p4,f_._p1,P_)_or_(_p4_`1_=_-_1_&_p4_`2_<_0_&_p1_`2_>_p4_`2_&_LE_f_._p4,f_._p1,P_)_) percases ( p4 `1 <> - 1 or ( p4 `1 = - 1 & p4 `2 >= 0 ) or ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 > p4 `2 ) ) by A66; caseA68: p4 `1 <> - 1 ; ::_thesis: LE f . p4,f . p1,P A69: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A70: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A71: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A72: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A73: f . p1 in P by A9, A69, A70, FUNCT_1:def_6; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then A74: f . p1 in Lower_Arc P by A71, A73; A75: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A76: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A70, A75, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; now__::_thesis:_(_(_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_LE_f_._p4,f_._p1,P_)_or_(_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_LE_f_._p4,f_._p1,P_)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_LE_f_._p4,f_._p1,P_)_) percases ( p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A67, A68, Th1; case p4 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: LE f . p4,f . p1,P then A77: p4 `2 = 1 by Th3; A78: f . p4 in P by A12, A69, A70, FUNCT_1:def_6; (f . p4) `2 >= 0 by A2, A77, Th69; then f . p4 in Upper_Arc P by A72, A78; hence LE f . p4,f . p1,P by A74, A76, JORDAN6:def_10; ::_thesis: verum end; case p4 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: LE f . p4,f . p1,P then A79: p4 `1 = 1 by Th1; A80: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A81: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A82: f . p4 in P by A12, A80, FUNCT_1:def_6; A83: f . p1 in P by A9, A80, A81, FUNCT_1:def_6; A84: (f . p1) `1 < 0 by A2, A21, A29, Th67; A85: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A86: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A87: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A81, A86, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A88: (f . p4) `1 >= 0 by A2, A79, Th68; now__::_thesis:_(_(_(f_._p4)_`2_>=_0_&_LE_f_._p4,f_._p1,P_)_or_(_(f_._p4)_`2_<_0_&_LE_f_._p4,f_._p1,P_)_) percases ( (f . p4) `2 >= 0 or (f . p4) `2 < 0 ) ; caseA89: (f . p4) `2 >= 0 ; ::_thesis: LE f . p4,f . p1,P A90: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A91: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A92: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A93: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A81, A92, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A94: f . p4 in Upper_Arc P by A82, A89, A91; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then f . p1 in Lower_Arc P by A83, A90; hence LE f . p4,f . p1,P by A93, A94, JORDAN6:def_10; ::_thesis: verum end; case (f . p4) `2 < 0 ; ::_thesis: LE f . p4,f . p1,P hence LE f . p4,f . p1,P by A5, A82, A83, A84, A85, A87, A88, JGRAPH_5:56; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; caseA95: p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ; ::_thesis: LE f . p4,f . p1,P then p4 `2 = - 1 by Th3; then A96: (f . p4) `2 < 0 by A2, Th69; A97: f . p4 in P by A12, A69, A70, FUNCT_1:def_6; A98: (f . p1) `2 <= 0 by A2, A21, A29, Th67; (f . p4) `1 >= (f . p1) `1 by A2, A20, A95, Th70; hence LE f . p4,f . p1,P by A5, A73, A76, A96, A97, A98, JGRAPH_5:56; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; caseA99: ( p4 `1 = - 1 & p4 `2 >= 0 ) ; ::_thesis: LE f . p4,f . p1,P A100: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A101: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A102: f . p4 in P by A12, A100, FUNCT_1:def_6; A103: f . p1 in P by A9, A100, A101, FUNCT_1:def_6; A104: (f . p4) `2 >= 0 by A2, A99, Th69; A105: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A106: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A107: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A108: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A101, A107, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A109: f . p4 in Upper_Arc P by A102, A104, A106; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then f . p1 in Lower_Arc P by A103, A105; hence LE f . p4,f . p1,P by A108, A109, JORDAN6:def_10; ::_thesis: verum end; caseA110: ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 > p4 `2 ) ; ::_thesis: LE f . p4,f . p1,P then A111: p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) by A13, Th2; A112: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A113: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A114: f . p4 in P by A12, A112, FUNCT_1:def_6; A115: f . p1 in P by A9, A112, A113, FUNCT_1:def_6; A116: (f . p1) `1 < 0 by A2, A21, A29, Th67; A117: (f . p1) `2 < 0 by A2, A21, A29, Th67; A118: (f . p4) `2 <= (f . p1) `2 by A2, A20, A29, A110, A111, Th71; (f . p4) `1 < 0 by A2, A110, Th68; hence LE f . p4,f . p1,P by A5, A114, A115, A116, A117, A118, JGRAPH_5:51; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; end; end; A119: rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } by Lm15; thus rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P proof thus rectangle ((- 1),1,(- 1),1) c= { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } c= rectangle ((- 1),1,(- 1),1) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rectangle ((- 1),1,(- 1),1) or x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ) assume x in rectangle ((- 1),1,(- 1),1) ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) ) by A119; hence x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ; ::_thesis: verum end; thus { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } c= rectangle ((- 1),1,(- 1),1) ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } or x in rectangle ((- 1),1,(- 1),1) ) assume x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ; ::_thesis: x in rectangle ((- 1),1,(- 1),1) then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) ) ; hence x in rectangle ((- 1),1,(- 1),1) by A119; ::_thesis: verum end; end; thus f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A58, A59, A60, JORDAN17:def_1; ::_thesis: verum end; end; end; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ; ::_thesis: verum end; caseA120: ( not p3 `2 < 0 or not p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; ::_thesis: ( rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } & f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) A121: now__::_thesis:_(_(_p3_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_(_(_p3_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p3_`2_)_or_p3_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p3_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p3_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p3_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p3_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_(_(_p3_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p3_`2_)_or_p3_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p3_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p3_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p3_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p3_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_(_(_p3_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p3_`2_)_or_p3_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p3_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p3_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p3_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p3_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_(_(_p3_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p3_`2_)_or_p3_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p3_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p3_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p3_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_) percases ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A11, Th63; case p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ; ::_thesis: ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A120, EUCLID:52; ::_thesis: verum end; case p3 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) ; ::_thesis: verum end; case p3 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) ; ::_thesis: verum end; caseA122: p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ; ::_thesis: ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) A123: W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; now__::_thesis:_not_p3_=_W-min_(rectangle_((-_1),1,(-_1),1)) assume A124: p3 = W-min (rectangle ((- 1),1,(- 1),1)) ; ::_thesis: contradiction then p3 `2 = - 1 by A123, EUCLID:52; hence contradiction by A120, A123, A124, RLTOPSP1:68; ::_thesis: verum end; hence ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A122; ::_thesis: verum end; end; end; then A125: LE |[(- 1),0]|,p3, rectangle ((- 1),1,(- 1),1) by A19, Th59; A126: (f . p2) `1 < 0 by A2, A30, A31, Th67; A127: (f . p2) `2 < 0 by A2, A30, A31, Th67; (f . p1) `2 <= (f . p2) `2 by A2, A20, A30, A33, Th71; then A128: LE f . p1,f . p2,P by A5, A22, A25, A32, A126, A127, JGRAPH_5:51; A129: LE f . p3,f . p4,P by A1, A2, A8, A17, A18, A125, Th66, RLTOPSP1:69; A130: now__::_thesis:_(_(_p4_`1_=_-_1_&_p4_`2_<_0_&_p1_`2_<=_p4_`2_&_contradiction_)_or_(_(_not_p4_`1_=_-_1_or_not_p4_`2_<_0_or_not_p1_`2_<=_p4_`2_)_&_LE_f_._p4,f_._p1,P_)_) percases ( ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 <= p4 `2 ) or not p4 `1 = - 1 or not p4 `2 < 0 or not p1 `2 <= p4 `2 ) ; caseA131: ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 <= p4 `2 ) ; ::_thesis: contradiction A132: |[(- 1),(- 1)]| `1 = - 1 by EUCLID:52; A133: |[(- 1),(- 1)]| `2 = - 1 by EUCLID:52; A134: |[(- 1),1]| `1 = - 1 by EUCLID:52; A135: |[(- 1),1]| `2 = 1 by EUCLID:52; - 1 <= p4 `2 by A12, Th19; then A136: p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) by A131, A132, A133, A134, A135, GOBOARD7:7; now__::_thesis:_(_(_p3_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p3_`2_&_contradiction_)_or_(_p3_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_contradiction_)_or_(_p3_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_contradiction_)_or_(_p3_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p3_<>_W-min_(rectangle_((-_1),1,(-_1),1))_&_contradiction_)_) percases ( ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) or p3 in LSeg (|[(- 1),1]|,|[1,1]|) or p3 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A121; caseA137: ( p3 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p3 `2 ) ; ::_thesis: contradiction then 0 <= p3 `2 by EUCLID:52; hence contradiction by A8, A131, A136, A137, Th55; ::_thesis: verum end; caseA138: p3 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: contradiction then LE p4,p3, rectangle ((- 1),1,(- 1),1) by A136, Th59; then p3 = p4 by A8, Th50, JORDAN6:57; hence contradiction by A131, A138, Th3; ::_thesis: verum end; caseA139: p3 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: contradiction then LE p4,p3, rectangle ((- 1),1,(- 1),1) by A136, Th59; then p3 = p4 by A8, Th50, JORDAN6:57; hence contradiction by A131, A139, Th1; ::_thesis: verum end; caseA140: ( p3 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p3 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ; ::_thesis: contradiction then LE p4,p3, rectangle ((- 1),1,(- 1),1) by A136, Th59; then A141: p3 = p4 by A8, Th50, JORDAN6:57; A142: p3 `2 = - 1 by A140, Th3; A143: W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; then A144: (W-min (rectangle ((- 1),1,(- 1),1))) `1 = - 1 by EUCLID:52; (W-min (rectangle ((- 1),1,(- 1),1))) `2 = - 1 by A143, EUCLID:52; hence contradiction by A131, A140, A141, A142, A144, TOPREAL3:6; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA145: ( not p4 `1 = - 1 or not p4 `2 < 0 or not p1 `2 <= p4 `2 ) ; ::_thesis: LE f . p4,f . p1,P A146: ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A12, Th63; now__::_thesis:_(_(_p4_`1_<>_-_1_&_LE_f_._p4,f_._p1,P_)_or_(_p4_`1_=_-_1_&_p4_`2_>=_0_&_LE_f_._p4,f_._p1,P_)_or_(_p4_`1_=_-_1_&_p4_`2_<_0_&_p1_`2_>_p4_`2_&_LE_f_._p4,f_._p1,P_)_) percases ( p4 `1 <> - 1 or ( p4 `1 = - 1 & p4 `2 >= 0 ) or ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 > p4 `2 ) ) by A145; caseA147: p4 `1 <> - 1 ; ::_thesis: LE f . p4,f . p1,P A148: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A149: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A150: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A151: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A152: f . p1 in P by A9, A148, A149, FUNCT_1:def_6; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then A153: f . p1 in Lower_Arc P by A150, A152; A154: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A155: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A149, A154, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; now__::_thesis:_(_(_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_LE_f_._p4,f_._p1,P_)_or_(_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_LE_f_._p4,f_._p1,P_)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_LE_f_._p4,f_._p1,P_)_) percases ( p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A146, A147, Th1; case p4 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: LE f . p4,f . p1,P then A156: p4 `2 = 1 by Th3; A157: f . p4 in P by A12, A148, A149, FUNCT_1:def_6; (f . p4) `2 >= 0 by A2, A156, Th69; then f . p4 in Upper_Arc P by A151, A157; hence LE f . p4,f . p1,P by A153, A155, JORDAN6:def_10; ::_thesis: verum end; case p4 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: LE f . p4,f . p1,P then A158: p4 `1 = 1 by Th1; A159: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A160: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A161: f . p4 in P by A12, A159, FUNCT_1:def_6; A162: f . p1 in P by A9, A159, A160, FUNCT_1:def_6; A163: (f . p1) `1 < 0 by A2, A21, A29, Th67; A164: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A165: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A166: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A160, A165, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A167: (f . p4) `1 >= 0 by A2, A158, Th68; now__::_thesis:_(_(_(f_._p4)_`2_>=_0_&_LE_f_._p4,f_._p1,P_)_or_(_(f_._p4)_`2_<_0_&_LE_f_._p4,f_._p1,P_)_) percases ( (f . p4) `2 >= 0 or (f . p4) `2 < 0 ) ; caseA168: (f . p4) `2 >= 0 ; ::_thesis: LE f . p4,f . p1,P A169: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A170: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A171: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A172: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A160, A171, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A173: f . p4 in Upper_Arc P by A161, A168, A170; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then f . p1 in Lower_Arc P by A162, A169; hence LE f . p4,f . p1,P by A172, A173, JORDAN6:def_10; ::_thesis: verum end; case (f . p4) `2 < 0 ; ::_thesis: LE f . p4,f . p1,P hence LE f . p4,f . p1,P by A5, A161, A162, A163, A164, A166, A167, JGRAPH_5:56; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; caseA174: p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ; ::_thesis: LE f . p4,f . p1,P then p4 `2 = - 1 by Th3; then A175: (f . p4) `2 < 0 by A2, Th69; A176: f . p4 in P by A12, A148, A149, FUNCT_1:def_6; A177: (f . p1) `2 <= 0 by A2, A21, A29, Th67; (f . p4) `1 >= (f . p1) `1 by A2, A20, A174, Th70; hence LE f . p4,f . p1,P by A5, A152, A155, A175, A176, A177, JGRAPH_5:56; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; caseA178: ( p4 `1 = - 1 & p4 `2 >= 0 ) ; ::_thesis: LE f . p4,f . p1,P A179: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A180: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A181: f . p4 in P by A12, A179, FUNCT_1:def_6; A182: f . p1 in P by A9, A179, A180, FUNCT_1:def_6; A183: (f . p4) `2 >= 0 by A2, A178, Th69; A184: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A185: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A186: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A187: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A180, A186, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A188: f . p4 in Upper_Arc P by A181, A183, A185; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then f . p1 in Lower_Arc P by A182, A184; hence LE f . p4,f . p1,P by A187, A188, JORDAN6:def_10; ::_thesis: verum end; caseA189: ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 > p4 `2 ) ; ::_thesis: LE f . p4,f . p1,P then A190: p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) by A13, Th2; A191: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A192: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A193: f . p4 in P by A12, A191, FUNCT_1:def_6; A194: f . p1 in P by A9, A191, A192, FUNCT_1:def_6; A195: (f . p1) `1 < 0 by A2, A21, A29, Th67; A196: (f . p1) `2 < 0 by A2, A21, A29, Th67; A197: (f . p4) `2 <= (f . p1) `2 by A2, A20, A29, A189, A190, Th71; (f . p4) `1 < 0 by A2, A189, Th68; hence LE f . p4,f . p1,P by A5, A193, A194, A195, A196, A197, JGRAPH_5:51; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; end; end; A198: rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } by Lm15; thus rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P proof thus rectangle ((- 1),1,(- 1),1) c= { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } c= rectangle ((- 1),1,(- 1),1) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rectangle ((- 1),1,(- 1),1) or x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ) assume x in rectangle ((- 1),1,(- 1),1) ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) ) by A198; hence x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ; ::_thesis: verum end; thus { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } c= rectangle ((- 1),1,(- 1),1) ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } or x in rectangle ((- 1),1,(- 1),1) ) assume x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ; ::_thesis: x in rectangle ((- 1),1,(- 1),1) then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) ) ; hence x in rectangle ((- 1),1,(- 1),1) by A198; ::_thesis: verum end; end; thus f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A128, A129, A130, JORDAN17:def_1; ::_thesis: verum end; end; end; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ; ::_thesis: verum end; caseA199: ( not p2 `2 < 0 or not p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ) ; ::_thesis: ( rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } & f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) A200: now__::_thesis:_(_(_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_(_(_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p2_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p2_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p2_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p2_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_(_(_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p2_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p2_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p2_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p2_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_(_(_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p2_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p2_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p2_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_or_(_p2_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_(_(_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p2_`2_)_or_p2_in_LSeg_(|[(-_1),1]|,|[1,1]|)_or_p2_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_or_(_p2_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p2_<>_W-min_(rectangle_((-_1),1,(-_1),1))_)_)_)_) percases ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A10, Th63; case p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) ; ::_thesis: ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A199, EUCLID:52; ::_thesis: verum end; case p2 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) ; ::_thesis: verum end; case p2 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) hence ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) ; ::_thesis: verum end; caseA201: p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ; ::_thesis: ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) A202: W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; now__::_thesis:_not_p2_=_W-min_(rectangle_((-_1),1,(-_1),1)) assume A203: p2 = W-min (rectangle ((- 1),1,(- 1),1)) ; ::_thesis: contradiction then p2 `2 = - 1 by A202, EUCLID:52; hence contradiction by A199, A202, A203, RLTOPSP1:68; ::_thesis: verum end; hence ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A201; ::_thesis: verum end; end; end; then A204: LE |[(- 1),0]|,p2, rectangle ((- 1),1,(- 1),1) by A19, Th59; then A205: LE f . p2,f . p3,P by A1, A2, A7, A17, A18, Th66, RLTOPSP1:69; LE |[(- 1),0]|,p3, rectangle ((- 1),1,(- 1),1) by A7, A204, Th50, JORDAN6:58; then A206: LE f . p3,f . p4,P by A1, A2, A8, A17, A18, Th66, RLTOPSP1:69; A207: now__::_thesis:_(_(_p4_`1_=_-_1_&_p4_`2_<_0_&_p1_`2_<=_p4_`2_&_contradiction_)_or_(_(_not_p4_`1_=_-_1_or_not_p4_`2_<_0_or_not_p1_`2_<=_p4_`2_)_&_LE_f_._p4,f_._p1,P_)_) percases ( ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 <= p4 `2 ) or not p4 `1 = - 1 or not p4 `2 < 0 or not p1 `2 <= p4 `2 ) ; caseA208: ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 <= p4 `2 ) ; ::_thesis: contradiction A209: |[(- 1),(- 1)]| `1 = - 1 by EUCLID:52; A210: |[(- 1),(- 1)]| `2 = - 1 by EUCLID:52; A211: |[(- 1),1]| `1 = - 1 by EUCLID:52; A212: |[(- 1),1]| `2 = 1 by EUCLID:52; - 1 <= p4 `2 by A12, Th19; then A213: p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) by A208, A209, A210, A211, A212, GOBOARD7:7; now__::_thesis:_(_(_p2_in_LSeg_(|[(-_1),(-_1)]|,|[(-_1),1]|)_&_|[(-_1),0]|_`2_<=_p2_`2_&_contradiction_)_or_(_p2_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_contradiction_)_or_(_p2_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_contradiction_)_or_(_p2_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_p2_<>_W-min_(rectangle_((-_1),1,(-_1),1))_&_contradiction_)_) percases ( ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) or p2 in LSeg (|[(- 1),1]|,|[1,1]|) or p2 in LSeg (|[1,1]|,|[1,(- 1)]|) or ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ) by A200; caseA214: ( p2 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) & |[(- 1),0]| `2 <= p2 `2 ) ; ::_thesis: contradiction then 0 <= p2 `2 by EUCLID:52; hence contradiction by A15, A208, A213, A214, Th55; ::_thesis: verum end; caseA215: p2 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: contradiction then LE p4,p2, rectangle ((- 1),1,(- 1),1) by A213, Th59; then p2 = p4 by A15, Th50, JORDAN6:57; hence contradiction by A208, A215, Th3; ::_thesis: verum end; caseA216: p2 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: contradiction then LE p4,p2, rectangle ((- 1),1,(- 1),1) by A213, Th59; then p2 = p4 by A15, Th50, JORDAN6:57; hence contradiction by A208, A216, Th1; ::_thesis: verum end; caseA217: ( p2 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p2 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ; ::_thesis: contradiction then LE p4,p2, rectangle ((- 1),1,(- 1),1) by A213, Th59; then A218: p2 = p4 by A15, Th50, JORDAN6:57; A219: p2 `2 = - 1 by A217, Th3; A220: W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; then A221: (W-min (rectangle ((- 1),1,(- 1),1))) `1 = - 1 by EUCLID:52; (W-min (rectangle ((- 1),1,(- 1),1))) `2 = - 1 by A220, EUCLID:52; hence contradiction by A208, A217, A218, A219, A221, TOPREAL3:6; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA222: ( not p4 `1 = - 1 or not p4 `2 < 0 or not p1 `2 <= p4 `2 ) ; ::_thesis: LE f . p4,f . p1,P A223: ( p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) or p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A12, Th63; now__::_thesis:_(_(_p4_`1_<>_-_1_&_LE_f_._p4,f_._p1,P_)_or_(_p4_`1_=_-_1_&_p4_`2_>=_0_&_LE_f_._p4,f_._p1,P_)_or_(_p4_`1_=_-_1_&_p4_`2_<_0_&_p1_`2_>_p4_`2_&_LE_f_._p4,f_._p1,P_)_) percases ( p4 `1 <> - 1 or ( p4 `1 = - 1 & p4 `2 >= 0 ) or ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 > p4 `2 ) ) by A222; caseA224: p4 `1 <> - 1 ; ::_thesis: LE f . p4,f . p1,P A225: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A226: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A227: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A228: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A229: f . p1 in P by A9, A225, A226, FUNCT_1:def_6; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then A230: f . p1 in Lower_Arc P by A227, A229; A231: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A232: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A226, A231, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; now__::_thesis:_(_(_p4_in_LSeg_(|[(-_1),1]|,|[1,1]|)_&_LE_f_._p4,f_._p1,P_)_or_(_p4_in_LSeg_(|[1,1]|,|[1,(-_1)]|)_&_LE_f_._p4,f_._p1,P_)_or_(_p4_in_LSeg_(|[1,(-_1)]|,|[(-_1),(-_1)]|)_&_LE_f_._p4,f_._p1,P_)_) percases ( p4 in LSeg (|[(- 1),1]|,|[1,1]|) or p4 in LSeg (|[1,1]|,|[1,(- 1)]|) or p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ) by A223, A224, Th1; case p4 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: LE f . p4,f . p1,P then A233: p4 `2 = 1 by Th3; A234: f . p4 in P by A12, A225, A226, FUNCT_1:def_6; (f . p4) `2 >= 0 by A2, A233, Th69; then f . p4 in Upper_Arc P by A228, A234; hence LE f . p4,f . p1,P by A230, A232, JORDAN6:def_10; ::_thesis: verum end; case p4 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: LE f . p4,f . p1,P then A235: p4 `1 = 1 by Th1; A236: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A237: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A238: f . p4 in P by A12, A236, FUNCT_1:def_6; A239: f . p1 in P by A9, A236, A237, FUNCT_1:def_6; A240: (f . p1) `1 < 0 by A2, A21, A29, Th67; A241: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A242: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A243: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A237, A242, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A244: (f . p4) `1 >= 0 by A2, A235, Th68; now__::_thesis:_(_(_(f_._p4)_`2_>=_0_&_LE_f_._p4,f_._p1,P_)_or_(_(f_._p4)_`2_<_0_&_LE_f_._p4,f_._p1,P_)_) percases ( (f . p4) `2 >= 0 or (f . p4) `2 < 0 ) ; caseA245: (f . p4) `2 >= 0 ; ::_thesis: LE f . p4,f . p1,P A246: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A247: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A248: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A249: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A237, A248, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A250: f . p4 in Upper_Arc P by A238, A245, A247; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then f . p1 in Lower_Arc P by A239, A246; hence LE f . p4,f . p1,P by A249, A250, JORDAN6:def_10; ::_thesis: verum end; case (f . p4) `2 < 0 ; ::_thesis: LE f . p4,f . p1,P hence LE f . p4,f . p1,P by A5, A238, A239, A240, A241, A243, A244, JGRAPH_5:56; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; caseA251: p4 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) ; ::_thesis: LE f . p4,f . p1,P then p4 `2 = - 1 by Th3; then A252: (f . p4) `2 < 0 by A2, Th69; A253: f . p4 in P by A12, A225, A226, FUNCT_1:def_6; A254: (f . p1) `2 <= 0 by A2, A21, A29, Th67; (f . p4) `1 >= (f . p1) `1 by A2, A20, A251, Th70; hence LE f . p4,f . p1,P by A5, A229, A232, A252, A253, A254, JGRAPH_5:56; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; caseA255: ( p4 `1 = - 1 & p4 `2 >= 0 ) ; ::_thesis: LE f . p4,f . p1,P A256: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A257: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A258: f . p4 in P by A12, A256, FUNCT_1:def_6; A259: f . p1 in P by A9, A256, A257, FUNCT_1:def_6; A260: (f . p4) `2 >= 0 by A2, A255, Th69; A261: (f . p1) `2 <= 0 by A2, A21, A29, Th67; A262: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A5, JGRAPH_5:34; A263: f . |[(- 1),0]| = W-min P by A2, A5, Th10, JGRAPH_5:29; A264: now__::_thesis:_not_f_._p1_=_W-min_P assume f . p1 = W-min P ; ::_thesis: contradiction then p1 = |[(- 1),0]| by A2, A257, A263, FUNCT_1:def_4; hence contradiction by A29, EUCLID:52; ::_thesis: verum end; A265: f . p4 in Upper_Arc P by A258, A260, A262; Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A5, JGRAPH_5:35; then f . p1 in Lower_Arc P by A259, A261; hence LE f . p4,f . p1,P by A264, A265, JORDAN6:def_10; ::_thesis: verum end; caseA266: ( p4 `1 = - 1 & p4 `2 < 0 & p1 `2 > p4 `2 ) ; ::_thesis: LE f . p4,f . p1,P then A267: p4 in LSeg (|[(- 1),(- 1)]|,|[(- 1),1]|) by A13, Th2; A268: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A5, Lm15, Th35, JGRAPH_3:23; A269: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A270: f . p4 in P by A12, A268, FUNCT_1:def_6; A271: f . p1 in P by A9, A268, A269, FUNCT_1:def_6; A272: (f . p1) `1 < 0 by A2, A21, A29, Th67; A273: (f . p1) `2 < 0 by A2, A21, A29, Th67; A274: (f . p4) `2 <= (f . p1) `2 by A2, A20, A29, A266, A267, Th71; (f . p4) `1 < 0 by A2, A266, Th68; hence LE f . p4,f . p1,P by A5, A270, A271, A272, A273, A274, JGRAPH_5:51; ::_thesis: verum end; end; end; hence LE f . p4,f . p1,P ; ::_thesis: verum end; end; end; A275: rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) } by Lm15; thus rectangle ((- 1),1,(- 1),1) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P proof thus rectangle ((- 1),1,(- 1),1) c= { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } c= rectangle ((- 1),1,(- 1),1) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rectangle ((- 1),1,(- 1),1) or x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ) assume x in rectangle ((- 1),1,(- 1),1) ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = 1 & - 1 <= p `1 & p `1 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `2 = - 1 & - 1 <= p `1 & p `1 <= 1 ) ) ) by A275; hence x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ; ::_thesis: verum end; thus { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } c= rectangle ((- 1),1,(- 1),1) ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } or x in rectangle ((- 1),1,(- 1),1) ) assume x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) } ; ::_thesis: x in rectangle ((- 1),1,(- 1),1) then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = - 1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) ) ; hence x in rectangle ((- 1),1,(- 1),1) by A275; ::_thesis: verum end; end; thus f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A205, A206, A207, JORDAN17:def_1; ::_thesis: verum end; end; end; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ; ::_thesis: verum end; end; end; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ; ::_thesis: verum end; caseA276: p1 in LSeg (|[(- 1),1]|,|[1,1]|) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P A277: |[(- 1),1]| in LSeg (|[(- 1),1]|,|[1,1]|) by RLTOPSP1:68; A278: |[(- 1),1]| `1 = - 1 by EUCLID:52; A279: |[(- 1),1]| `2 = 1 by EUCLID:52; - 1 <= p1 `1 by A276, Th3; then A280: LE |[(- 1),1]|,p1, rectangle ((- 1),1,(- 1),1) by A276, A277, A278, Th60; then A281: LE f . p1,f . p2,P by A1, A2, A6, A279, Th66, RLTOPSP1:68; A282: LE |[(- 1),1]|,p2, rectangle ((- 1),1,(- 1),1) by A6, A280, Th50, JORDAN6:58; then A283: LE f . p2,f . p3,P by A1, A2, A7, A279, Th66, RLTOPSP1:68; LE |[(- 1),1]|,p3, rectangle ((- 1),1,(- 1),1) by A7, A282, Th50, JORDAN6:58; then LE f . p3,f . p4,P by A1, A2, A8, A279, Th66, RLTOPSP1:68; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A281, A283, JORDAN17:def_1; ::_thesis: verum end; caseA284: p1 in LSeg (|[1,1]|,|[1,(- 1)]|) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P A285: |[(- 1),1]| in LSeg (|[(- 1),1]|,|[1,1]|) by RLTOPSP1:68; A286: |[(- 1),1]| `1 = - 1 by EUCLID:52; A287: |[(- 1),1]| `2 = 1 by EUCLID:52; A288: |[1,1]| in LSeg (|[1,1]|,|[1,(- 1)]|) by RLTOPSP1:68; A289: |[1,1]| in LSeg (|[(- 1),1]|,|[1,1]|) by RLTOPSP1:68; A290: |[1,1]| `1 = 1 by EUCLID:52; A291: |[1,1]| `2 = 1 by EUCLID:52; A292: LE |[(- 1),1]|,|[1,1]|, rectangle ((- 1),1,(- 1),1) by A285, A286, A289, A290, Th60; p1 `2 <= 1 by A284, Th1; then LE |[1,1]|,p1, rectangle ((- 1),1,(- 1),1) by A284, A288, A291, Th61; then A293: LE |[(- 1),1]|,p1, rectangle ((- 1),1,(- 1),1) by A292, Th50, JORDAN6:58; then A294: LE f . p1,f . p2,P by A1, A2, A6, A287, Th66, RLTOPSP1:68; A295: LE |[(- 1),1]|,p2, rectangle ((- 1),1,(- 1),1) by A6, A293, Th50, JORDAN6:58; then A296: LE f . p2,f . p3,P by A1, A2, A7, A287, Th66, RLTOPSP1:68; LE |[(- 1),1]|,p3, rectangle ((- 1),1,(- 1),1) by A7, A295, Th50, JORDAN6:58; then LE f . p3,f . p4,P by A1, A2, A8, A287, Th66, RLTOPSP1:68; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A294, A296, JORDAN17:def_1; ::_thesis: verum end; caseA297: ( p1 in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) & p1 <> W-min (rectangle ((- 1),1,(- 1),1)) ) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P A298: |[(- 1),1]| in LSeg (|[(- 1),1]|,|[1,1]|) by RLTOPSP1:68; A299: |[(- 1),1]| `1 = - 1 by EUCLID:52; A300: |[(- 1),1]| `2 = 1 by EUCLID:52; A301: |[1,1]| in LSeg (|[(- 1),1]|,|[1,1]|) by RLTOPSP1:68; |[1,1]| `1 = 1 by EUCLID:52; then A302: LE |[(- 1),1]|,|[1,1]|, rectangle ((- 1),1,(- 1),1) by A298, A299, A301, Th60; A303: |[1,(- 1)]| in LSeg (|[1,1]|,|[1,(- 1)]|) by RLTOPSP1:68; A304: |[1,(- 1)]| in LSeg (|[1,(- 1)]|,|[(- 1),(- 1)]|) by RLTOPSP1:68; A305: |[1,(- 1)]| `1 = 1 by EUCLID:52; LE |[1,1]|,|[1,(- 1)]|, rectangle ((- 1),1,(- 1),1) by A301, A303, Th60; then A306: LE |[(- 1),1]|,|[1,(- 1)]|, rectangle ((- 1),1,(- 1),1) by A302, Th50, JORDAN6:58; W-min (rectangle ((- 1),1,(- 1),1)) = |[(- 1),(- 1)]| by Th46; then A307: (W-min (rectangle ((- 1),1,(- 1),1))) `1 = - 1 by EUCLID:52; p1 `1 <= 1 by A297, Th3; then LE |[1,(- 1)]|,p1, rectangle ((- 1),1,(- 1),1) by A297, A304, A305, A307, Th62; then A308: LE |[(- 1),1]|,p1, rectangle ((- 1),1,(- 1),1) by A306, Th50, JORDAN6:58; then A309: LE f . p1,f . p2,P by A1, A2, A6, A300, Th66, RLTOPSP1:68; A310: LE |[(- 1),1]|,p2, rectangle ((- 1),1,(- 1),1) by A6, A308, Th50, JORDAN6:58; then A311: LE f . p2,f . p3,P by A1, A2, A7, A300, Th66, RLTOPSP1:68; LE |[(- 1),1]|,p3, rectangle ((- 1),1,(- 1),1) by A7, A310, Th50, JORDAN6:58; then LE f . p3,f . p4,P by A1, A2, A8, A300, Th66, RLTOPSP1:68; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A309, A311, JORDAN17:def_1; ::_thesis: verum end; end; end; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ; ::_thesis: verum end; end; theorem Th73: :: JGRAPH_6:73 for p1, p2 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & not LE p1,p2,P holds LE p2,p1,P proof let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & not LE p1,p2,P holds LE p2,p1,P let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & p1 in P & p2 in P & not LE p1,p2,P implies LE p2,p1,P ) assume that A1: P is being_simple_closed_curve and A2: p1 in P and A3: p2 in P and A4: not LE p1,p2,P ; ::_thesis: LE p2,p1,P A5: P = (Upper_Arc P) \/ (Lower_Arc P) by A1, JORDAN6:def_9; A6: not p1 = W-min P by A1, A3, A4, JORDAN7:3; percases ( ( p1 in Upper_Arc P & p2 in Upper_Arc P ) or ( p1 in Upper_Arc P & p2 in Lower_Arc P ) or ( p1 in Lower_Arc P & p2 in Upper_Arc P ) or ( p1 in Lower_Arc P & p2 in Lower_Arc P ) ) by A2, A3, A5, XBOOLE_0:def_3; supposeA7: ( p1 in Upper_Arc P & p2 in Upper_Arc P ) ; ::_thesis: LE p2,p1,P A8: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, JORDAN6:def_8; set q1 = W-min P; set q2 = E-max P; set Q = Upper_Arc P; now__::_thesis:_(_(_p1_<>_p2_&_LE_p2,p1,P_)_or_(_p1_=_p2_&_LE_p2,p1,P_)_) percases ( p1 <> p2 or p1 = p2 ) ; caseA9: p1 <> p2 ; ::_thesis: LE p2,p1,P now__::_thesis:_(_(_LE_p1,p2,_Upper_Arc_P,_W-min_P,_E-max_P_&_not_LE_p2,p1,_Upper_Arc_P,_W-min_P,_E-max_P_&_contradiction_)_or_(_LE_p2,p1,_Upper_Arc_P,_W-min_P,_E-max_P_&_not_LE_p1,p2,_Upper_Arc_P,_W-min_P,_E-max_P_&_LE_p2,p1,P_)_) percases ( ( LE p1,p2, Upper_Arc P, W-min P, E-max P & not LE p2,p1, Upper_Arc P, W-min P, E-max P ) or ( LE p2,p1, Upper_Arc P, W-min P, E-max P & not LE p1,p2, Upper_Arc P, W-min P, E-max P ) ) by A7, A8, A9, JORDAN5C:14; case ( LE p1,p2, Upper_Arc P, W-min P, E-max P & not LE p2,p1, Upper_Arc P, W-min P, E-max P ) ; ::_thesis: contradiction hence contradiction by A4, A7, JORDAN6:def_10; ::_thesis: verum end; case ( LE p2,p1, Upper_Arc P, W-min P, E-max P & not LE p1,p2, Upper_Arc P, W-min P, E-max P ) ; ::_thesis: LE p2,p1,P hence LE p2,p1,P by A7, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p2,p1,P ; ::_thesis: verum end; case p1 = p2 ; ::_thesis: LE p2,p1,P hence LE p2,p1,P by A1, A2, JORDAN6:56; ::_thesis: verum end; end; end; hence LE p2,p1,P ; ::_thesis: verum end; supposeA10: ( p1 in Upper_Arc P & p2 in Lower_Arc P ) ; ::_thesis: LE p2,p1,P now__::_thesis:_(_(_p2_=_W-min_P_&_LE_p2,p1,P_)_or_(_p2_<>_W-min_P_&_contradiction_)_) percases ( p2 = W-min P or p2 <> W-min P ) ; case p2 = W-min P ; ::_thesis: LE p2,p1,P hence LE p2,p1,P by A1, A2, JORDAN7:3; ::_thesis: verum end; case p2 <> W-min P ; ::_thesis: contradiction hence contradiction by A4, A10, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p2,p1,P ; ::_thesis: verum end; suppose ( p1 in Lower_Arc P & p2 in Upper_Arc P ) ; ::_thesis: LE p2,p1,P hence LE p2,p1,P by A6, JORDAN6:def_10; ::_thesis: verum end; supposeA11: ( p1 in Lower_Arc P & p2 in Lower_Arc P ) ; ::_thesis: LE p2,p1,P A12: Lower_Arc P is_an_arc_of E-max P, W-min P by A1, JORDAN6:50; set q2 = W-min P; set q1 = E-max P; set Q = Lower_Arc P; now__::_thesis:_(_(_p1_<>_p2_&_LE_p2,p1,P_)_or_(_p1_=_p2_&_LE_p2,p1,P_)_) percases ( p1 <> p2 or p1 = p2 ) ; caseA13: p1 <> p2 ; ::_thesis: LE p2,p1,P now__::_thesis:_(_(_LE_p1,p2,_Lower_Arc_P,_E-max_P,_W-min_P_&_not_LE_p2,p1,_Lower_Arc_P,_E-max_P,_W-min_P_&_LE_p2,p1,P_)_or_(_LE_p2,p1,_Lower_Arc_P,_E-max_P,_W-min_P_&_not_LE_p1,p2,_Lower_Arc_P,_E-max_P,_W-min_P_&_LE_p2,p1,P_)_) percases ( ( LE p1,p2, Lower_Arc P, E-max P, W-min P & not LE p2,p1, Lower_Arc P, E-max P, W-min P ) or ( LE p2,p1, Lower_Arc P, E-max P, W-min P & not LE p1,p2, Lower_Arc P, E-max P, W-min P ) ) by A11, A12, A13, JORDAN5C:14; caseA14: ( LE p1,p2, Lower_Arc P, E-max P, W-min P & not LE p2,p1, Lower_Arc P, E-max P, W-min P ) ; ::_thesis: LE p2,p1,P now__::_thesis:_(_(_p2_=_W-min_P_&_LE_p2,p1,P_)_or_(_p2_<>_W-min_P_&_contradiction_)_) percases ( p2 = W-min P or p2 <> W-min P ) ; case p2 = W-min P ; ::_thesis: LE p2,p1,P hence LE p2,p1,P by A1, A2, JORDAN7:3; ::_thesis: verum end; case p2 <> W-min P ; ::_thesis: contradiction hence contradiction by A4, A11, A14, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p2,p1,P ; ::_thesis: verum end; case ( LE p2,p1, Lower_Arc P, E-max P, W-min P & not LE p1,p2, Lower_Arc P, E-max P, W-min P ) ; ::_thesis: LE p2,p1,P hence LE p2,p1,P by A6, A11, JORDAN6:def_10; ::_thesis: verum end; end; end; hence LE p2,p1,P ; ::_thesis: verum end; case p1 = p2 ; ::_thesis: LE p2,p1,P hence LE p2,p1,P by A1, A2, JORDAN6:56; ::_thesis: verum end; end; end; hence LE p2,p1,P ; ::_thesis: verum end; end; end; theorem :: JGRAPH_6:74 for p1, p2, p3 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P & not ( LE p1,p2,P & LE p2,p3,P ) & not ( LE p1,p3,P & LE p3,p2,P ) & not ( LE p2,p1,P & LE p1,p3,P ) & not ( LE p2,p3,P & LE p3,p1,P ) & not ( LE p3,p1,P & LE p1,p2,P ) holds ( LE p3,p2,P & LE p2,p1,P ) by Th73; theorem :: JGRAPH_6:75 for p1, p2, p3 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P & LE p2,p3,P & not LE p1,p2,P & not ( LE p2,p1,P & LE p1,p3,P ) holds LE p3,p1,P by Th73; theorem :: JGRAPH_6:76 for p1, p2, p3, p4 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P & p4 in P & LE p2,p3,P & LE p3,p4,P & not LE p1,p2,P & not ( LE p2,p1,P & LE p1,p3,P ) & not ( LE p3,p1,P & LE p1,p4,P ) holds LE p4,p1,P by Th73; theorem Th77: :: JGRAPH_6:77 for p1, p2, p3, p4 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & LE f . p1,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p4,P holds p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) proof let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & LE f . p1,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p4,P holds p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ & LE f . p1,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p4,P holds p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( P = circle (0,0,1) & f = Sq_Circ & LE f . p1,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p4,P implies p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ) set K = rectangle ((- 1),1,(- 1),1); assume that A1: P = circle (0,0,1) and A2: f = Sq_Circ and A3: LE f . p1,f . p2,P and A4: LE f . p2,f . p3,P and A5: LE f . p3,f . p4,P ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) A6: rectangle ((- 1),1,(- 1),1) is being_simple_closed_curve by Th50; A7: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } by A1, Th24; then A8: LE f . p1,f . p3,P by A3, A4, JGRAPH_3:26, JORDAN6:58; A9: LE f . p2,f . p4,P by A4, A5, A7, JGRAPH_3:26, JORDAN6:58; A10: f .: (rectangle ((- 1),1,(- 1),1)) = P by A2, A7, Lm15, Th35, JGRAPH_3:23; A11: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A12: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } by A1, Th24; then A13: P is being_simple_closed_curve by JGRAPH_3:26; then f . p1 in P by A3, JORDAN7:5; then ex x1 being set st ( x1 in dom f & x1 in rectangle ((- 1),1,(- 1),1) & f . p1 = f . x1 ) by A10, FUNCT_1:def_6; then A14: p1 in rectangle ((- 1),1,(- 1),1) by A2, A11, FUNCT_1:def_4; f . p2 in P by A3, A13, JORDAN7:5; then ex x2 being set st ( x2 in dom f & x2 in rectangle ((- 1),1,(- 1),1) & f . p2 = f . x2 ) by A10, FUNCT_1:def_6; then A15: p2 in rectangle ((- 1),1,(- 1),1) by A2, A11, FUNCT_1:def_4; f . p3 in P by A4, A13, JORDAN7:5; then ex x3 being set st ( x3 in dom f & x3 in rectangle ((- 1),1,(- 1),1) & f . p3 = f . x3 ) by A10, FUNCT_1:def_6; then A16: p3 in rectangle ((- 1),1,(- 1),1) by A2, A11, FUNCT_1:def_4; f . p4 in P by A5, A13, JORDAN7:5; then ex x4 being set st ( x4 in dom f & x4 in rectangle ((- 1),1,(- 1),1) & f . p4 = f . x4 ) by A10, FUNCT_1:def_6; then A17: p4 in rectangle ((- 1),1,(- 1),1) by A2, A11, FUNCT_1:def_4; now__::_thesis:_p1,p2,p3,p4_are_in_this_order_on_rectangle_((-_1),1,(-_1),1) assume A18: not p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: contradiction A19: now__::_thesis:_not_p1,p2,p4,p3_are_in_this_order_on_rectangle_((-_1),1,(-_1),1) assume A20: p1,p2,p4,p3 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: contradiction now__::_thesis:_(_(_LE_p1,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p3,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p2,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p1,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p4,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p2,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p3,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p4,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_) percases ( ( LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p3, rectangle ((- 1),1,(- 1),1) ) or ( LE p2,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p1, rectangle ((- 1),1,(- 1),1) ) or ( LE p4,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p2, rectangle ((- 1),1,(- 1),1) ) or ( LE p3,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p4, rectangle ((- 1),1,(- 1),1) ) ) by A20, JORDAN17:def_1; caseA21: ( LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p3, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p1,f . p2,f . p4,f . p3 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p1,f . p2,P & LE f . p2,f . p4,P & LE f . p4,f . p3,P ) or ( LE f . p2,f . p4,P & LE f . p4,f . p3,P & LE f . p3,f . p1,P ) or ( LE f . p4,f . p3,P & LE f . p3,f . p1,P & LE f . p1,f . p2,P ) or ( LE f . p3,f . p1,P & LE f . p1,f . p2,P & LE f . p2,f . p4,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p4 or f . p3 = f . p1 ) by A5, A8, A12, JGRAPH_3:26, JORDAN6:57; then A22: ( p3 = p4 or p3 = p1 ) by A2, A11, FUNCT_1:def_4; LE p1,p4, rectangle ((- 1),1,(- 1),1) by A21, Th50, JORDAN6:58; then p1 = p4 by A18, A20, A21, A22, Th50, JORDAN6:57; hence contradiction by A18, A20, A22; ::_thesis: verum end; caseA23: ( LE p2,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p1, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p2,f . p4,f . p3,f . p1 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p2,f . p4,P & LE f . p4,f . p3,P & LE f . p3,f . p1,P ) or ( LE f . p4,f . p3,P & LE f . p3,f . p1,P & LE f . p1,f . p2,P ) or ( LE f . p3,f . p1,P & LE f . p1,f . p2,P & LE f . p2,f . p4,P ) or ( LE f . p1,f . p2,P & LE f . p2,f . p4,P & LE f . p4,f . p3,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p4 or LE f . p3,f . p2,P ) by A5, A13, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p4 or f . p3 = f . p2 ) by A4, A12, JGRAPH_3:26, JORDAN6:57; then A24: ( p3 = p4 or p3 = p2 ) by A2, A11, FUNCT_1:def_4; then p4 = p2 by A18, A20, A23, Th50, JORDAN6:57; hence contradiction by A18, A20, A24; ::_thesis: verum end; caseA25: ( LE p4,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p2, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p4,f . p3,f . p1,f . p2 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p4,f . p3,P & LE f . p3,f . p1,P & LE f . p1,f . p2,P ) or ( LE f . p3,f . p1,P & LE f . p1,f . p2,P & LE f . p2,f . p4,P ) or ( LE f . p1,f . p2,P & LE f . p2,f . p4,P & LE f . p4,f . p3,P ) or ( LE f . p2,f . p4,P & LE f . p4,f . p3,P & LE f . p3,f . p1,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p4 or LE f . p3,f . p2,P ) by A5, A13, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p4 or f . p3 = f . p2 ) by A4, A12, JGRAPH_3:26, JORDAN6:57; then A26: ( p3 = p4 or p3 = p2 ) by A2, A11, FUNCT_1:def_4; then p3 = p1 by A18, A20, A25, Th50, JORDAN6:57; hence contradiction by A6, A18, A20, A26, JORDAN17:12; ::_thesis: verum end; caseA27: ( LE p3,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p4, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p3,f . p1,f . p2,f . p4 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p4,f . p3,P & LE f . p3,f . p1,P & LE f . p1,f . p2,P ) or ( LE f . p3,f . p1,P & LE f . p1,f . p2,P & LE f . p2,f . p4,P ) or ( LE f . p1,f . p2,P & LE f . p2,f . p4,P & LE f . p4,f . p3,P ) or ( LE f . p2,f . p4,P & LE f . p4,f . p3,P & LE f . p3,f . p1,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p4 or LE f . p3,f . p2,P ) by A5, A13, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p4 or f . p3 = f . p2 ) by A4, A12, JGRAPH_3:26, JORDAN6:57; then A28: ( p3 = p4 or p3 = p2 ) by A2, A11, FUNCT_1:def_4; then p3 = p1 by A18, A20, A27, Th50, JORDAN6:57; hence contradiction by A6, A18, A20, A28, JORDAN17:12; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; A29: now__::_thesis:_not_p1,p3,p4,p2_are_in_this_order_on_rectangle_((-_1),1,(-_1),1) assume A30: p1,p3,p4,p2 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: contradiction now__::_thesis:_(_(_LE_p1,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p2,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p3,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p1,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p4,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p3,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p2,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p4,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_) percases ( ( LE p1,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p2, rectangle ((- 1),1,(- 1),1) ) or ( LE p3,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p1, rectangle ((- 1),1,(- 1),1) ) or ( LE p4,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p3, rectangle ((- 1),1,(- 1),1) ) or ( LE p2,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) ) ) by A30, JORDAN17:def_1; case ( LE p1,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p2, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p1,f . p3,f . p4,f . p2 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p1,f . p3,P & LE f . p3,f . p4,P & LE f . p4,f . p2,P ) or ( LE f . p3,f . p4,P & LE f . p4,f . p2,P & LE f . p2,f . p1,P ) or ( LE f . p4,f . p2,P & LE f . p2,f . p1,P & LE f . p1,f . p3,P ) or ( LE f . p2,f . p1,P & LE f . p1,f . p3,P & LE f . p3,f . p4,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p2 or f . p2 = f . p1 ) by A3, A9, A12, JGRAPH_3:26, JORDAN6:57; then A31: ( p4 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then ( f . p3 = f . p2 or f . p4 = f . p1 ) by A4, A5, A6, A12, A18, A30, JGRAPH_3:26, JORDAN17:12, JORDAN6:57; then ( p3 = p2 or p4 = p1 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A18, A30, A31, JORDAN17:12; ::_thesis: verum end; case ( LE p3,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p1, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p3,f . p4,f . p2,f . p1 are_in_this_order_on P by A1, A2, Th72; then A32: ( ( LE f . p1,f . p3,P & LE f . p3,f . p4,P & LE f . p4,f . p2,P ) or ( LE f . p3,f . p4,P & LE f . p4,f . p2,P & LE f . p2,f . p1,P ) or ( LE f . p4,f . p2,P & LE f . p2,f . p1,P & LE f . p1,f . p3,P ) or ( LE f . p2,f . p1,P & LE f . p1,f . p3,P & LE f . p3,f . p4,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p2 or f . p2 = f . p1 ) by A3, A9, A12, JGRAPH_3:26, JORDAN6:57; then A33: ( p4 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; ( f . p2 = f . p1 or LE f . p3,f . p2,P ) by A3, A13, A32, JORDAN6:57, JORDAN6:58; then ( f . p2 = f . p1 or f . p3 = f . p2 ) by A4, A12, JGRAPH_3:26, JORDAN6:57; then ( p2 = p1 or p3 = p2 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A18, A30, A33, JORDAN17:12; ::_thesis: verum end; case ( LE p4,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p3, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p4,f . p2,f . p1,f . p3 are_in_this_order_on P by A1, A2, Th72; then A34: ( ( LE f . p1,f . p3,P & LE f . p3,f . p4,P & LE f . p4,f . p2,P ) or ( LE f . p3,f . p4,P & LE f . p4,f . p2,P & LE f . p2,f . p1,P ) or ( LE f . p4,f . p2,P & LE f . p2,f . p1,P & LE f . p1,f . p3,P ) or ( LE f . p2,f . p1,P & LE f . p1,f . p3,P & LE f . p3,f . p4,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p2 or f . p2 = f . p1 ) by A3, A9, A12, JGRAPH_3:26, JORDAN6:57; then A35: ( p4 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; ( f . p2 = f . p1 or LE f . p3,f . p2,P ) by A3, A13, A34, JORDAN6:57, JORDAN6:58; then ( f . p2 = f . p1 or f . p3 = f . p2 ) by A4, A12, JGRAPH_3:26, JORDAN6:57; then ( p2 = p1 or p3 = p2 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A18, A30, A35, JORDAN17:12; ::_thesis: verum end; caseA36: ( LE p2,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p2,f . p1,f . p3,f . p4 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p1,f . p3,P & LE f . p3,f . p4,P & LE f . p4,f . p2,P ) or ( LE f . p3,f . p4,P & LE f . p4,f . p2,P & LE f . p2,f . p1,P ) or ( LE f . p4,f . p2,P & LE f . p2,f . p1,P & LE f . p1,f . p3,P ) or ( LE f . p2,f . p1,P & LE f . p1,f . p3,P & LE f . p3,f . p4,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p2 or f . p2 = f . p1 ) by A3, A9, A12, JGRAPH_3:26, JORDAN6:57; then A37: ( p4 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; LE p2,p3, rectangle ((- 1),1,(- 1),1) by A36, Th50, JORDAN6:58; then p2 = p3 by A6, A18, A30, A36, A37, JORDAN17:12, JORDAN6:57; hence contradiction by A6, A18, A30, A37, JORDAN17:12; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; now__::_thesis:_(_(_p1,p2,p4,p3_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_p1,p3,p2,p4_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_p1,p3,p4,p2_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_p1,p4,p2,p3_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_p1,p4,p3,p2_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_) percases ( p1,p2,p4,p3 are_in_this_order_on rectangle ((- 1),1,(- 1),1) or p1,p3,p2,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) or p1,p3,p4,p2 are_in_this_order_on rectangle ((- 1),1,(- 1),1) or p1,p4,p2,p3 are_in_this_order_on rectangle ((- 1),1,(- 1),1) or p1,p4,p3,p2 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ) by A6, A14, A15, A16, A17, A18, JORDAN17:27; case p1,p2,p4,p3 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: contradiction hence contradiction by A19; ::_thesis: verum end; caseA38: p1,p3,p2,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: contradiction now__::_thesis:_(_(_LE_p1,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p4,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p3,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p1,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p2,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p3,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p4,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p2,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_) percases ( ( LE p1,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p4, rectangle ((- 1),1,(- 1),1) ) or ( LE p3,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p1, rectangle ((- 1),1,(- 1),1) ) or ( LE p2,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p3, rectangle ((- 1),1,(- 1),1) ) or ( LE p4,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p2, rectangle ((- 1),1,(- 1),1) ) ) by A38, JORDAN17:def_1; caseA39: ( LE p1,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p4, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p1,f . p3,f . p2,f . p4 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p1,f . p3,P & LE f . p3,f . p2,P & LE f . p2,f . p4,P ) or ( LE f . p3,f . p2,P & LE f . p2,f . p4,P & LE f . p4,f . p1,P ) or ( LE f . p2,f . p4,P & LE f . p4,f . p1,P & LE f . p1,f . p3,P ) or ( LE f . p4,f . p1,P & LE f . p1,f . p3,P & LE f . p3,f . p2,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p2 or LE f . p2,f . p1,P ) by A4, A13, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p2 or f . p2 = f . p1 ) by A3, A12, JGRAPH_3:26, JORDAN6:57; then A40: ( p3 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then p3 = p1 by A18, A38, A39, Th50, JORDAN6:57; hence contradiction by A18, A38, A40; ::_thesis: verum end; caseA41: ( LE p3,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p1, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p3,f . p2,f . p4,f . p1 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p1,f . p3,P & LE f . p3,f . p2,P & LE f . p2,f . p4,P ) or ( LE f . p3,f . p2,P & LE f . p2,f . p4,P & LE f . p4,f . p1,P ) or ( LE f . p2,f . p4,P & LE f . p4,f . p1,P & LE f . p1,f . p3,P ) or ( LE f . p4,f . p1,P & LE f . p1,f . p3,P & LE f . p3,f . p2,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p2 or LE f . p2,f . p1,P ) by A4, A13, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p2 or f . p2 = f . p1 ) by A3, A12, JGRAPH_3:26, JORDAN6:57; then A42: ( p3 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then p4 = p1 by A18, A38, A41, Th50, JORDAN6:57; hence contradiction by A6, A18, A38, A42, JORDAN17:12; ::_thesis: verum end; caseA43: ( LE p2,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p3, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p2,f . p4,f . p1,f . p3 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p1,f . p3,P & LE f . p3,f . p2,P & LE f . p2,f . p4,P ) or ( LE f . p3,f . p2,P & LE f . p2,f . p4,P & LE f . p4,f . p1,P ) or ( LE f . p2,f . p4,P & LE f . p4,f . p1,P & LE f . p1,f . p3,P ) or ( LE f . p4,f . p1,P & LE f . p1,f . p3,P & LE f . p3,f . p2,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p2 or LE f . p2,f . p1,P ) by A4, A13, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p2 or f . p2 = f . p1 ) by A3, A12, JGRAPH_3:26, JORDAN6:57; then A44: ( p3 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then p4 = p1 by A18, A38, A43, Th50, JORDAN6:57; hence contradiction by A6, A18, A38, A44, JORDAN17:12; ::_thesis: verum end; caseA45: ( LE p4,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p2, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p4,f . p1,f . p3,f . p2 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p1,f . p3,P & LE f . p3,f . p2,P & LE f . p2,f . p4,P ) or ( LE f . p3,f . p2,P & LE f . p2,f . p4,P & LE f . p4,f . p1,P ) or ( LE f . p2,f . p4,P & LE f . p4,f . p1,P & LE f . p1,f . p3,P ) or ( LE f . p4,f . p1,P & LE f . p1,f . p3,P & LE f . p3,f . p2,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p2 or LE f . p2,f . p1,P ) by A4, A13, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p2 or f . p2 = f . p1 ) by A3, A12, JGRAPH_3:26, JORDAN6:57; then A46: ( p3 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then p3 = p1 by A18, A38, A45, Th50, JORDAN6:57; hence contradiction by A18, A38, A46; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; case p1,p3,p4,p2 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: contradiction hence contradiction by A29; ::_thesis: verum end; caseA47: p1,p4,p2,p3 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: contradiction now__::_thesis:_(_(_LE_p1,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p3,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p4,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p1,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p2,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p4,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p3,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p2,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_) percases ( ( LE p1,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) ) or ( LE p4,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p1, rectangle ((- 1),1,(- 1),1) ) or ( LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p4, rectangle ((- 1),1,(- 1),1) ) or ( LE p3,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p2, rectangle ((- 1),1,(- 1),1) ) ) by A47, JORDAN17:def_1; caseA48: ( LE p1,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p1,f . p4,f . p2,f . p3 are_in_this_order_on P by A1, A2, Th72; then A49: ( ( LE f . p1,f . p4,P & LE f . p4,f . p2,P & LE f . p2,f . p3,P ) or ( LE f . p4,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p1,P ) or ( LE f . p2,f . p3,P & LE f . p3,f . p1,P & LE f . p1,f . p4,P ) or ( LE f . p3,f . p1,P & LE f . p1,f . p4,P & LE f . p4,f . p2,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p2 or LE f . p2,f . p1,P ) by A9, A13, JORDAN6:57, JORDAN6:58; then ( f . p4 = f . p2 or f . p2 = f . p1 ) by A3, A12, JGRAPH_3:26, JORDAN6:57; then A50: ( p4 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then A51: p4 = p2 by A48, Th50, JORDAN6:57; ( f . p3 = f . p1 or LE f . p4,f . p3,P ) by A8, A13, A49, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p1 or f . p4 = f . p3 ) by A5, A12, JGRAPH_3:26, JORDAN6:57; then A52: ( p3 = p1 or p4 = p3 ) by A2, A11, FUNCT_1:def_4; then p1 = p2 by A18, A47, A48, A50, Th50, JORDAN6:57; hence contradiction by A18, A47, A51, A52; ::_thesis: verum end; caseA53: ( LE p4,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p1, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p4,f . p2,f . p3,f . p1 are_in_this_order_on P by A1, A2, Th72; then A54: ( ( LE f . p1,f . p4,P & LE f . p4,f . p2,P & LE f . p2,f . p3,P ) or ( LE f . p4,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p1,P ) or ( LE f . p2,f . p3,P & LE f . p3,f . p1,P & LE f . p1,f . p4,P ) or ( LE f . p3,f . p1,P & LE f . p1,f . p4,P & LE f . p4,f . p2,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p2 or LE f . p2,f . p1,P ) by A9, A13, JORDAN6:57, JORDAN6:58; then ( f . p4 = f . p2 or f . p2 = f . p1 ) by A3, A12, JGRAPH_3:26, JORDAN6:57; then ( p4 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then A55: ( p4 = p2 or ( p2 = p1 & p3 = p1 ) ) by A53, Th50, JORDAN6:57; ( f . p3 = f . p1 or LE f . p4,f . p3,P ) by A8, A13, A54, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p1 or f . p4 = f . p3 ) by A5, A12, JGRAPH_3:26, JORDAN6:57; then ( p3 = p1 or p4 = p3 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A29, A47, A55, JORDAN17:12; ::_thesis: verum end; caseA56: ( LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p4, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p2,f . p3,f . p1,f . p4 are_in_this_order_on P by A1, A2, Th72; then A57: ( ( LE f . p1,f . p4,P & LE f . p4,f . p2,P & LE f . p2,f . p3,P ) or ( LE f . p4,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p1,P ) or ( LE f . p2,f . p3,P & LE f . p3,f . p1,P & LE f . p1,f . p4,P ) or ( LE f . p3,f . p1,P & LE f . p1,f . p4,P & LE f . p4,f . p2,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p2 or LE f . p2,f . p1,P ) by A9, A13, JORDAN6:57, JORDAN6:58; then ( f . p4 = f . p2 or f . p2 = f . p1 ) by A3, A12, JGRAPH_3:26, JORDAN6:57; then ( p4 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then A58: ( p4 = p2 or ( p2 = p1 & p3 = p1 ) ) by A56, Th50, JORDAN6:57; ( f . p3 = f . p1 or LE f . p4,f . p3,P ) by A8, A13, A57, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p1 or f . p4 = f . p3 ) by A5, A12, JGRAPH_3:26, JORDAN6:57; then ( p3 = p1 or p4 = p3 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A29, A47, A58, JORDAN17:12; ::_thesis: verum end; caseA59: ( LE p3,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p2, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p3,f . p1,f . p4,f . p2 are_in_this_order_on P by A1, A2, Th72; then A60: ( ( LE f . p1,f . p4,P & LE f . p4,f . p2,P & LE f . p2,f . p3,P ) or ( LE f . p4,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p1,P ) or ( LE f . p2,f . p3,P & LE f . p3,f . p1,P & LE f . p1,f . p4,P ) or ( LE f . p3,f . p1,P & LE f . p1,f . p4,P & LE f . p4,f . p2,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p2 or LE f . p2,f . p1,P ) by A9, A13, JORDAN6:57, JORDAN6:58; then ( f . p4 = f . p2 or f . p2 = f . p1 ) by A3, A12, JGRAPH_3:26, JORDAN6:57; then ( p4 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then A61: p4 = p2 by A59, Th50, JORDAN6:57; ( f . p3 = f . p1 or LE f . p4,f . p3,P ) by A8, A13, A60, JORDAN6:57, JORDAN6:58; then ( f . p3 = f . p1 or f . p4 = f . p3 ) by A5, A12, JGRAPH_3:26, JORDAN6:57; then ( p3 = p1 or p4 = p3 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A29, A47, A61, JORDAN17:12; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; caseA62: p1,p4,p3,p2 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: contradiction now__::_thesis:_(_(_LE_p1,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p2,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p4,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p1,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p3,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p4,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_or_(_LE_p2,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p3,_rectangle_((-_1),1,(-_1),1)_&_contradiction_)_) percases ( ( LE p1,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p2, rectangle ((- 1),1,(- 1),1) ) or ( LE p4,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p1, rectangle ((- 1),1,(- 1),1) ) or ( LE p3,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p4, rectangle ((- 1),1,(- 1),1) ) or ( LE p2,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p3, rectangle ((- 1),1,(- 1),1) ) ) by A62, JORDAN17:def_1; caseA63: ( LE p1,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p2, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p1,f . p4,f . p3,f . p2 are_in_this_order_on P by A1, A2, Th72; then A64: ( ( LE f . p1,f . p4,P & LE f . p4,f . p3,P & LE f . p3,f . p2,P ) or ( LE f . p4,f . p3,P & LE f . p3,f . p2,P & LE f . p2,f . p1,P ) or ( LE f . p3,f . p2,P & LE f . p2,f . p1,P & LE f . p1,f . p4,P ) or ( LE f . p2,f . p1,P & LE f . p1,f . p4,P & LE f . p4,f . p3,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p2 or f . p2 = f . p1 ) by A3, A4, A12, JGRAPH_3:26, JORDAN6:57; then A65: ( p3 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; LE p1,p3, rectangle ((- 1),1,(- 1),1) by A63, Th50, JORDAN6:58; then A66: p3 = p2 by A63, A65, Th50, JORDAN6:57; ( f . p4 = f . p3 or f . p2 = f . p1 ) by A3, A5, A12, A64, JGRAPH_3:26, JORDAN6:57; then ( p4 = p3 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; then ( p4 = p3 or p2,p3,p4,p1 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ) by A6, A62, A66, JORDAN17:12; hence contradiction by A6, A18, A62, A65, JORDAN17:12; ::_thesis: verum end; case ( LE p4,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p1, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p4,f . p3,f . p2,f . p1 are_in_this_order_on P by A1, A2, Th72; then A67: ( ( LE f . p1,f . p4,P & LE f . p4,f . p3,P & LE f . p3,f . p2,P ) or ( LE f . p4,f . p3,P & LE f . p3,f . p2,P & LE f . p2,f . p1,P ) or ( LE f . p3,f . p2,P & LE f . p2,f . p1,P & LE f . p1,f . p4,P ) or ( LE f . p2,f . p1,P & LE f . p1,f . p4,P & LE f . p4,f . p3,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p2 or f . p2 = f . p1 ) by A3, A4, A12, JGRAPH_3:26, JORDAN6:57; then A68: ( p3 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; ( f . p4 = f . p3 or f . p2 = f . p1 ) by A3, A5, A12, A67, JGRAPH_3:26, JORDAN6:57; then ( p4 = p3 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A19, A62, A68, JORDAN17:12; ::_thesis: verum end; case ( LE p3,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p4, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p3,f . p2,f . p1,f . p4 are_in_this_order_on P by A1, A2, Th72; then ( ( LE f . p1,f . p4,P & LE f . p4,f . p3,P & LE f . p3,f . p2,P ) or ( LE f . p4,f . p3,P & LE f . p3,f . p2,P & LE f . p2,f . p1,P ) or ( LE f . p3,f . p2,P & LE f . p2,f . p1,P & LE f . p1,f . p4,P ) or ( LE f . p2,f . p1,P & LE f . p1,f . p4,P & LE f . p4,f . p3,P ) ) by JORDAN17:def_1; then ( f . p4 = f . p3 or f . p2 = f . p1 ) by A3, A5, A12, JGRAPH_3:26, JORDAN6:57; then ( p4 = p3 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A19, A29, A62, JORDAN17:12; ::_thesis: verum end; caseA69: ( LE p2,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p3, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: contradiction then f . p2,f . p1,f . p4,f . p3 are_in_this_order_on P by A1, A2, Th72; then A70: ( ( LE f . p1,f . p4,P & LE f . p4,f . p3,P & LE f . p3,f . p2,P ) or ( LE f . p4,f . p3,P & LE f . p3,f . p2,P & LE f . p2,f . p1,P ) or ( LE f . p3,f . p2,P & LE f . p2,f . p1,P & LE f . p1,f . p4,P ) or ( LE f . p2,f . p1,P & LE f . p1,f . p4,P & LE f . p4,f . p3,P ) ) by JORDAN17:def_1; then ( f . p3 = f . p2 or f . p2 = f . p1 ) by A3, A4, A12, JGRAPH_3:26, JORDAN6:57; then A71: ( p3 = p2 or p2 = p1 ) by A2, A11, FUNCT_1:def_4; LE p1,p3, rectangle ((- 1),1,(- 1),1) by A69, Th50, JORDAN6:58; then A72: p1 = p2 by A69, A71, Th50, JORDAN6:57; ( f . p4 = f . p3 or f . p2 = f . p3 ) by A4, A5, A12, A70, JGRAPH_3:26, JORDAN6:57; then ( p4 = p3 or p2 = p3 ) by A2, A11, FUNCT_1:def_4; hence contradiction by A6, A29, A62, A72, JORDAN17:12; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; hence p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: verum end; theorem Th78: :: JGRAPH_6:78 for p1, p2, p3, p4 being Point of (TOP-REAL 2) for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ holds ( p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) iff f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) proof let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ holds ( p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) iff f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st P = circle (0,0,1) & f = Sq_Circ holds ( p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) iff f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( P = circle (0,0,1) & f = Sq_Circ implies ( p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) iff f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) ) set K = rectangle ((- 1),1,(- 1),1); assume that A1: P = circle (0,0,1) and A2: f = Sq_Circ ; ::_thesis: ( p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) iff f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) A3: rectangle ((- 1),1,(- 1),1) is being_simple_closed_curve by Th50; circle (0,0,1) = { p5 where p5 is Point of (TOP-REAL 2) : |.p5.| = 1 } by Th24; then A4: P is being_simple_closed_curve by A1, JGRAPH_3:26; thus ( p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) implies f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ) ::_thesis: ( f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P implies p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ) proof assume A5: p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P now__::_thesis:_(_(_LE_p1,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p4,_rectangle_((-_1),1,(-_1),1)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_LE_p2,p3,_rectangle_((-_1),1,(-_1),1)_&_LE_p3,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p1,_rectangle_((-_1),1,(-_1),1)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_LE_p3,p4,_rectangle_((-_1),1,(-_1),1)_&_LE_p4,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p2,_rectangle_((-_1),1,(-_1),1)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_or_(_LE_p4,p1,_rectangle_((-_1),1,(-_1),1)_&_LE_p1,p2,_rectangle_((-_1),1,(-_1),1)_&_LE_p2,p3,_rectangle_((-_1),1,(-_1),1)_&_f_._p1,f_._p2,f_._p3,f_._p4_are_in_this_order_on_P_)_) percases ( ( LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) ) or ( LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p1, rectangle ((- 1),1,(- 1),1) ) or ( LE p3,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p2, rectangle ((- 1),1,(- 1),1) ) or ( LE p4,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) ) ) by A5, JORDAN17:def_1; case ( LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A1, A2, Th72; ::_thesis: verum end; case ( LE p2,p3, rectangle ((- 1),1,(- 1),1) & LE p3,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p1, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A1, A2, A4, Th72, JORDAN17:12; ::_thesis: verum end; case ( LE p3,p4, rectangle ((- 1),1,(- 1),1) & LE p4,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p2, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A1, A2, A4, Th72, JORDAN17:11; ::_thesis: verum end; case ( LE p4,p1, rectangle ((- 1),1,(- 1),1) & LE p1,p2, rectangle ((- 1),1,(- 1),1) & LE p2,p3, rectangle ((- 1),1,(- 1),1) ) ; ::_thesis: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P by A1, A2, A4, Th72, JORDAN17:10; ::_thesis: verum end; end; end; hence f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ; ::_thesis: verum end; thus ( f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P implies p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ) ::_thesis: verum proof assume A6: f . p1,f . p2,f . p3,f . p4 are_in_this_order_on P ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) now__::_thesis:_(_(_LE_f_._p1,f_._p2,P_&_LE_f_._p2,f_._p3,P_&_LE_f_._p3,f_._p4,P_&_p1,p2,p3,p4_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_)_or_(_LE_f_._p2,f_._p3,P_&_LE_f_._p3,f_._p4,P_&_LE_f_._p4,f_._p1,P_&_p1,p2,p3,p4_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_)_or_(_LE_f_._p3,f_._p4,P_&_LE_f_._p4,f_._p1,P_&_LE_f_._p1,f_._p2,P_&_p1,p2,p3,p4_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_)_or_(_LE_f_._p4,f_._p1,P_&_LE_f_._p1,f_._p2,P_&_LE_f_._p2,f_._p3,P_&_p1,p2,p3,p4_are_in_this_order_on_rectangle_((-_1),1,(-_1),1)_)_) percases ( ( LE f . p1,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p4,P ) or ( LE f . p2,f . p3,P & LE f . p3,f . p4,P & LE f . p4,f . p1,P ) or ( LE f . p3,f . p4,P & LE f . p4,f . p1,P & LE f . p1,f . p2,P ) or ( LE f . p4,f . p1,P & LE f . p1,f . p2,P & LE f . p2,f . p3,P ) ) by A6, JORDAN17:def_1; case ( LE f . p1,f . p2,P & LE f . p2,f . p3,P & LE f . p3,f . p4,P ) ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) hence p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A1, A2, Th77; ::_thesis: verum end; case ( LE f . p2,f . p3,P & LE f . p3,f . p4,P & LE f . p4,f . p1,P ) ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) then p2,p3,p4,p1 are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A1, A2, Th77; hence p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A3, JORDAN17:12; ::_thesis: verum end; case ( LE f . p3,f . p4,P & LE f . p4,f . p1,P & LE f . p1,f . p2,P ) ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) then p3,p4,p1,p2 are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A1, A2, Th77; hence p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A3, JORDAN17:11; ::_thesis: verum end; case ( LE f . p4,f . p1,P & LE f . p1,f . p2,P & LE f . p2,f . p3,P ) ; ::_thesis: p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) then p4,p1,p2,p3 are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A1, A2, Th77; hence p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A3, JORDAN17:10; ::_thesis: verum end; end; end; hence p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: verum end; end; theorem :: JGRAPH_6:79 for p1, p2, p3, p4 being Point of (TOP-REAL 2) st p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) holds for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= closed_inside_of_rectangle ((- 1),1,(- 1),1) & rng g c= closed_inside_of_rectangle ((- 1),1,(- 1),1) holds rng f meets rng g proof let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: ( p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= closed_inside_of_rectangle ((- 1),1,(- 1),1) & rng g c= closed_inside_of_rectangle ((- 1),1,(- 1),1) holds rng f meets rng g ) set K = rectangle ((- 1),1,(- 1),1); set K0 = closed_inside_of_rectangle ((- 1),1,(- 1),1); assume A1: p1,p2,p3,p4 are_in_this_order_on rectangle ((- 1),1,(- 1),1) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= closed_inside_of_rectangle ((- 1),1,(- 1),1) & rng g c= closed_inside_of_rectangle ((- 1),1,(- 1),1) holds rng f meets rng g reconsider j = 1 as real non negative number ; reconsider P = circle (0,0,j) as non empty compact Subset of (TOP-REAL 2) ; A2: P = { p6 where p6 is Point of (TOP-REAL 2) : |.p6.| = 1 } by Th24; thus for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= closed_inside_of_rectangle ((- 1),1,(- 1),1) & rng g c= closed_inside_of_rectangle ((- 1),1,(- 1),1) holds rng f meets rng g ::_thesis: verum proof let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= closed_inside_of_rectangle ((- 1),1,(- 1),1) & rng g c= closed_inside_of_rectangle ((- 1),1,(- 1),1) implies rng f meets rng g ) assume that A3: ( f is continuous & f is one-to-one ) and A4: ( g is continuous & g is one-to-one ) and A5: f . 0 = p1 and A6: f . 1 = p3 and A7: g . 0 = p2 and A8: g . 1 = p4 and A9: rng f c= closed_inside_of_rectangle ((- 1),1,(- 1),1) and A10: rng g c= closed_inside_of_rectangle ((- 1),1,(- 1),1) ; ::_thesis: rng f meets rng g reconsider s = Sq_Circ as Function of (TOP-REAL 2),(TOP-REAL 2) ; A11: dom s = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; reconsider f1 = s * f as Function of I[01],(TOP-REAL 2) ; reconsider g1 = s * g as Function of I[01],(TOP-REAL 2) ; s is being_homeomorphism by JGRAPH_3:43; then A12: s is continuous by TOPS_2:def_5; A13: dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then 0 in dom f by XXREAL_1:1; then A14: f1 . 0 = Sq_Circ . p1 by A5, FUNCT_1:13; 1 in dom f by A13, XXREAL_1:1; then A15: f1 . 1 = Sq_Circ . p3 by A6, FUNCT_1:13; A16: dom g = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then 0 in dom g by XXREAL_1:1; then A17: g1 . 0 = Sq_Circ . p2 by A7, FUNCT_1:13; 1 in dom g by A16, XXREAL_1:1; then A18: g1 . 1 = Sq_Circ . p4 by A8, FUNCT_1:13; defpred S1[ Point of (TOP-REAL 2)] means |.$1.| <= 1; { p8 where p8 is Point of (TOP-REAL 2) : S1[p8] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider C0 = { p8 where p8 is Point of (TOP-REAL 2) : |.p8.| <= 1 } as Subset of (TOP-REAL 2) ; A19: s .: (closed_inside_of_rectangle ((- 1),1,(- 1),1)) = C0 by Th27; A20: rng f1 c= C0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f1 or y in C0 ) assume y in rng f1 ; ::_thesis: y in C0 then consider x being set such that A21: x in dom f1 and A22: y = f1 . x by FUNCT_1:def_3; A23: x in dom f by A21, FUNCT_1:11; A24: f . x in dom s by A21, FUNCT_1:11; f . x in rng f by A23, FUNCT_1:3; then s . (f . x) in s .: (closed_inside_of_rectangle ((- 1),1,(- 1),1)) by A9, A24, FUNCT_1:def_6; hence y in C0 by A19, A21, A22, FUNCT_1:12; ::_thesis: verum end; A25: rng g1 c= C0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g1 or y in C0 ) assume y in rng g1 ; ::_thesis: y in C0 then consider x being set such that A26: x in dom g1 and A27: y = g1 . x by FUNCT_1:def_3; A28: x in dom g by A26, FUNCT_1:11; A29: g . x in dom s by A26, FUNCT_1:11; g . x in rng g by A28, FUNCT_1:3; then s . (g . x) in s .: (closed_inside_of_rectangle ((- 1),1,(- 1),1)) by A10, A29, FUNCT_1:def_6; hence y in C0 by A19, A26, A27, FUNCT_1:12; ::_thesis: verum end; reconsider q1 = s . p1, q2 = s . p2, q3 = s . p3, q4 = s . p4 as Point of (TOP-REAL 2) ; q1,q2,q3,q4 are_in_this_order_on P by A1, Th78; then rng f1 meets rng g1 by A2, A3, A4, A12, A14, A15, A17, A18, A20, A25, Th18; then consider y being set such that A30: y in rng f1 and A31: y in rng g1 by XBOOLE_0:3; consider x1 being set such that A32: x1 in dom f1 and A33: y = f1 . x1 by A30, FUNCT_1:def_3; consider x2 being set such that A34: x2 in dom g1 and A35: y = g1 . x2 by A31, FUNCT_1:def_3; dom f1 c= dom f by RELAT_1:25; then A36: f . x1 in rng f by A32, FUNCT_1:3; dom g1 c= dom g by RELAT_1:25; then A37: g . x2 in rng g by A34, FUNCT_1:3; y = Sq_Circ . (f . x1) by A32, A33, FUNCT_1:12; then A38: (Sq_Circ ") . y = f . x1 by A11, A36, FUNCT_1:32; x1 in dom f by A32, FUNCT_1:11; then A39: f . x1 in rng f by FUNCT_1:def_3; y = Sq_Circ . (g . x2) by A34, A35, FUNCT_1:12; then A40: (Sq_Circ ") . y = g . x2 by A11, A37, FUNCT_1:32; x2 in dom g by A34, FUNCT_1:11; then g . x2 in rng g by FUNCT_1:def_3; hence rng f meets rng g by A38, A39, A40, XBOOLE_0:3; ::_thesis: verum end; end;