:: JGRAPH_4 semantic presentation begin Lm1: for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds |.p.| > 0 proof let p be Point of (TOP-REAL 2); ::_thesis: ( p <> 0. (TOP-REAL 2) implies |.p.| > 0 ) assume p <> 0. (TOP-REAL 2) ; ::_thesis: |.p.| > 0 then |.p.| <> 0 by TOPRNS_1:24; hence |.p.| > 0 ; ::_thesis: verum end; theorem Th1: :: JGRAPH_4:1 for X being non empty TopStruct for g being Function of X,R^1 for B being Subset of X for a being real number st g is continuous & B = { p where p is Point of X : g /. p > a } holds B is open proof let X be non empty TopStruct ; ::_thesis: for g being Function of X,R^1 for B being Subset of X for a being real number st g is continuous & B = { p where p is Point of X : g /. p > a } holds B is open let g be Function of X,R^1; ::_thesis: for B being Subset of X for a being real number st g is continuous & B = { p where p is Point of X : g /. p > a } holds B is open let B be Subset of X; ::_thesis: for a being real number st g is continuous & B = { p where p is Point of X : g /. p > a } holds B is open let a be real number ; ::_thesis: ( g is continuous & B = { p where p is Point of X : g /. p > a } implies B is open ) assume that A1: g is continuous and A2: B = { p where p is Point of X : g /. p > a } ; ::_thesis: B is open { r where r is Real : r > a } c= the carrier of R^1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { r where r is Real : r > a } or x in the carrier of R^1 ) assume x in { r where r is Real : r > a } ; ::_thesis: x in the carrier of R^1 then ex r being Real st ( r = x & r > a ) ; hence x in the carrier of R^1 by TOPMETR:17; ::_thesis: verum end; then reconsider D = { r where r is Real : r > a } as Subset of R^1 ; A3: g " D c= B proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g " D or x in B ) assume A4: x in g " D ; ::_thesis: x in B then reconsider p = x as Point of X ; g . x in D by A4, FUNCT_1:def_7; then A5: ex r being Real st ( r = g . x & r > a ) ; g /. p = g . p ; hence x in B by A2, A5; ::_thesis: verum end; A6: ( [#] R^1 <> {} & D is open ) by JORDAN2B:25; B c= g " D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in B or x in g " D ) assume x in B ; ::_thesis: x in g " D then consider p being Point of X such that A7: p = x and A8: g /. p > a by A2; g /. p is Real by XREAL_0:def_1; then ( dom g = the carrier of X & g . x in D ) by A7, A8, FUNCT_2:def_1; hence x in g " D by A7, FUNCT_1:def_7; ::_thesis: verum end; then B = g " D by A3, XBOOLE_0:def_10; hence B is open by A1, A6, TOPS_2:43; ::_thesis: verum end; theorem Th2: :: JGRAPH_4:2 for X being non empty TopStruct for g being Function of X,R^1 for B being Subset of X for a being Real st g is continuous & B = { p where p is Point of X : g /. p < a } holds B is open proof let X be non empty TopStruct ; ::_thesis: for g being Function of X,R^1 for B being Subset of X for a being Real st g is continuous & B = { p where p is Point of X : g /. p < a } holds B is open let g be Function of X,R^1; ::_thesis: for B being Subset of X for a being Real st g is continuous & B = { p where p is Point of X : g /. p < a } holds B is open let B be Subset of X; ::_thesis: for a being Real st g is continuous & B = { p where p is Point of X : g /. p < a } holds B is open let a be Real; ::_thesis: ( g is continuous & B = { p where p is Point of X : g /. p < a } implies B is open ) assume that A1: g is continuous and A2: B = { p where p is Point of X : g /. p < a } ; ::_thesis: B is open { r where r is Real : r < a } c= the carrier of R^1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { r where r is Real : r < a } or x in the carrier of R^1 ) assume x in { r where r is Real : r < a } ; ::_thesis: x in the carrier of R^1 then ex r being Real st ( r = x & r < a ) ; hence x in the carrier of R^1 by TOPMETR:17; ::_thesis: verum end; then reconsider D = { r where r is Real : r < a } as Subset of R^1 ; A3: g " D c= B proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g " D or x in B ) assume A4: x in g " D ; ::_thesis: x in B then reconsider p = x as Point of X ; g . x in D by A4, FUNCT_1:def_7; then A5: ex r being Real st ( r = g . x & r < a ) ; g /. p = g . p ; hence x in B by A2, A5; ::_thesis: verum end; A6: ( [#] R^1 <> {} & D is open ) by JORDAN2B:24; B c= g " D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in B or x in g " D ) assume x in B ; ::_thesis: x in g " D then consider p being Point of X such that A7: p = x and A8: g /. p < a by A2; g /. p is Real by XREAL_0:def_1; then ( dom g = the carrier of X & g . x in D ) by A7, A8, FUNCT_2:def_1; hence x in g " D by A7, FUNCT_1:def_7; ::_thesis: verum end; then B = g " D by A3, XBOOLE_0:def_10; hence B is open by A1, A6, TOPS_2:43; ::_thesis: verum end; theorem Th3: :: JGRAPH_4:3 for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f is continuous & f is one-to-one & rng f = [#] (TOP-REAL 2) & ( for p2 being Point of (TOP-REAL 2) ex K being non empty compact Subset of (TOP-REAL 2) st ( K = f .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & f . p2 in V2 ) ) ) holds f is being_homeomorphism proof let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & rng f = [#] (TOP-REAL 2) & ( for p2 being Point of (TOP-REAL 2) ex K being non empty compact Subset of (TOP-REAL 2) st ( K = f .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & f . p2 in V2 ) ) ) implies f is being_homeomorphism ) assume that A1: ( f is continuous & f is one-to-one ) and A2: rng f = [#] (TOP-REAL 2) and A3: for p2 being Point of (TOP-REAL 2) ex K being non empty compact Subset of (TOP-REAL 2) st ( K = f .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & f . p2 in V2 ) ) ; ::_thesis: f is being_homeomorphism reconsider g = f " as Function of (TOP-REAL 2),(TOP-REAL 2) by A1, A2, FUNCT_2:25; A4: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; for p being Point of (TOP-REAL 2) for V being Subset of (TOP-REAL 2) st g . p in V & V is open holds ex W being Subset of (TOP-REAL 2) st ( p in W & W is open & g .: W c= V ) proof let p be Point of (TOP-REAL 2); ::_thesis: for V being Subset of (TOP-REAL 2) st g . p in V & V is open holds ex W being Subset of (TOP-REAL 2) st ( p in W & W is open & g .: W c= V ) let V be Subset of (TOP-REAL 2); ::_thesis: ( g . p in V & V is open implies ex W being Subset of (TOP-REAL 2) st ( p in W & W is open & g .: W c= V ) ) assume that A5: g . p in V and A6: V is open ; ::_thesis: ex W being Subset of (TOP-REAL 2) st ( p in W & W is open & g .: W c= V ) consider K being non empty compact Subset of (TOP-REAL 2) such that A7: K = f .: K and A8: ex V2 being Subset of (TOP-REAL 2) st ( g . p in V2 & V2 is open & V2 c= K & f . (g . p) in V2 ) by A3; consider V2 being Subset of (TOP-REAL 2) such that A9: g . p in V2 and A10: V2 is open and A11: V2 c= K and A12: f . (g . p) in V2 by A8; A13: dom (f | K) = (dom f) /\ K by RELAT_1:61 .= K by A4, XBOOLE_1:28 ; A14: g . p in V /\ V2 by A5, A9, XBOOLE_0:def_4; the carrier of ((TOP-REAL 2) | K) = K by PRE_TOPC:8; then reconsider R = (V /\ V2) /\ K as Subset of ((TOP-REAL 2) | K) by XBOOLE_1:17; A15: R = (V /\ V2) /\ ([#] ((TOP-REAL 2) | K)) by PRE_TOPC:def_5; V /\ V2 is open by A6, A10, TOPS_1:11; then A16: R is open by A15, TOPS_2:24; A17: p in V2 by A1, A2, A12, FUNCT_1:35; then reconsider q = p as Point of ((TOP-REAL 2) | K) by A11, PRE_TOPC:8; A18: rng (f | K) c= the carrier of (TOP-REAL 2) ; dom (f | K) = (dom f) /\ K by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K by FUNCT_2:def_1 .= K by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K) by PRE_TOPC:8 ; then reconsider h = f | K as Function of ((TOP-REAL 2) | K),(TOP-REAL 2) by A18, FUNCT_2:2; A19: h is one-to-one by A1, FUNCT_1:52; A20: K = (f | K) .: K by A7, RELAT_1:129 .= rng (f | K) by A13, RELAT_1:113 ; then consider f1 being Function of ((TOP-REAL 2) | K),((TOP-REAL 2) | K) such that A21: h = f1 and A22: f1 is being_homeomorphism by A1, A19, JGRAPH_1:46, TOPMETR:7; A23: rng f1 = [#] ((TOP-REAL 2) | K) by A22, TOPS_2:def_5; A24: f1 is onto by A23, FUNCT_2:def_3; ( dom (f1 ") = rng f1 & rng (f1 ") = dom f1 ) by A19, A21, FUNCT_1:33; then reconsider g1 = f1 " as Function of ((TOP-REAL 2) | K),((TOP-REAL 2) | K) by A23, FUNCT_2:2; g1 = f1 " by A19, A21, A24, TOPS_2:def_4; then A25: g1 is continuous by A22, TOPS_2:def_5; A26: f1 . (g . p) = f . (g . p) by A9, A11, A21, FUNCT_1:49 .= p by A1, A2, FUNCT_1:35 ; A27: dom f1 = (dom f) /\ K by A21, RELAT_1:61 .= K by A4, XBOOLE_1:28 ; rng f1 = dom (f1 ") by A19, A21, FUNCT_1:33; then A28: (f1 ") . p in rng (f1 ") by A11, A17, A20, A21, FUNCT_1:3; A29: rng (f1 ") = dom f1 by A19, A21, FUNCT_1:33; f1 . ((f1 ") . p) = p by A11, A17, A19, A20, A21, FUNCT_1:35; then (f1 ") . p = g . p by A8, A19, A21, A26, A27, A29, A28, FUNCT_1:def_4; then (f1 ") . p in R by A9, A11, A14, XBOOLE_0:def_4; then consider W3 being Subset of ((TOP-REAL 2) | K) such that A30: q in W3 and A31: W3 is open and A32: (f1 ") .: W3 c= R by A16, A25, JGRAPH_2:10; R = V /\ (V2 /\ K) by XBOOLE_1:16; then A33: R c= V by XBOOLE_1:17; consider W5 being Subset of (TOP-REAL 2) such that A34: W5 is open and A35: W3 = W5 /\ ([#] ((TOP-REAL 2) | K)) by A31, TOPS_2:24; reconsider W4 = W5 /\ V2 as Subset of (TOP-REAL 2) ; p in W5 by A30, A35, XBOOLE_0:def_4; then A36: p in W4 by A17, XBOOLE_0:def_4; A37: dom f1 = the carrier of ((TOP-REAL 2) | K) by FUNCT_2:def_1; A38: (f ") .: W3 c= R proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (f ") .: W3 or y in R ) assume y in (f ") .: W3 ; ::_thesis: y in R then consider x being set such that A39: x in dom (f ") and A40: x in W3 and A41: y = (f ") . x by FUNCT_1:def_6; A42: x in rng f by A1, A39, FUNCT_1:33; then A43: y in dom f by A1, A41, FUNCT_1:32; A44: f . y = x by A1, A41, A42, FUNCT_1:32; the carrier of ((TOP-REAL 2) | K) = K by PRE_TOPC:8; then ex z2 being set st ( z2 in dom f & z2 in K & f . y = f . z2 ) by A7, A40, A44, FUNCT_1:def_6; then A45: y in K by A1, A43, FUNCT_1:def_4; then A46: y in the carrier of ((TOP-REAL 2) | K) by PRE_TOPC:8; A47: dom (f1 ") = the carrier of ((TOP-REAL 2) | K) by A19, A21, A23, FUNCT_1:33; f1 . y = x by A21, A44, A45, FUNCT_1:49; then y = (f1 ") . x by A19, A21, A37, A46, FUNCT_1:32; then y in (f1 ") .: W3 by A40, A47, FUNCT_1:def_6; hence y in R by A32; ::_thesis: verum end; W4 = W5 /\ (V2 /\ K) by A11, XBOOLE_1:28 .= (W5 /\ K) /\ V2 by XBOOLE_1:16 .= W3 /\ V2 by A35, PRE_TOPC:def_5 ; then A48: g .: W4 c= (g .: W3) /\ (g .: V2) by RELAT_1:121; (g .: W3) /\ (g .: V2) c= g .: W3 by XBOOLE_1:17; then g .: W4 c= g .: W3 by A48, XBOOLE_1:1; then A49: g .: W4 c= R by A38, XBOOLE_1:1; W4 is open by A10, A34, TOPS_1:11; hence ex W being Subset of (TOP-REAL 2) st ( p in W & W is open & g .: W c= V ) by A36, A49, A33, XBOOLE_1:1; ::_thesis: verum end; then A50: g is continuous by JGRAPH_2:10; f is onto by A2, FUNCT_2:def_3; then g = f " by A1, TOPS_2:def_4; hence f is being_homeomorphism by A1, A2, A4, A50, TOPS_2:def_5; ::_thesis: verum end; theorem Th4: :: JGRAPH_4:4 for X being non empty TopSpace for f1, f2 being Function of X,R^1 for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = ((r1 / r2) - a) / b ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = ((r1 / r2) - a) / b ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = ((r1 / r2) - a) / b ) & g is continuous ) let a, b be real number ; ::_thesis: ( f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = ((r1 / r2) - a) / b ) & g is continuous ) ) assume that A1: ( f1 is continuous & f2 is continuous ) and A2: b <> 0 and A3: for q being Point of X holds f2 . q <> 0 ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = ((r1 / r2) - a) / b ) & g is continuous ) consider g3 being Function of X,R^1 such that A4: for p being Point of X for r1, r0 being real number st f1 . p = r1 & f2 . p = r0 holds g3 . p = r1 / r0 and A5: g3 is continuous by A1, A3, JGRAPH_2:27; consider g1 being Function of X,R^1 such that A6: for p being Point of X holds ( g1 . p = b & g1 is continuous ) by JGRAPH_2:20; consider g2 being Function of X,R^1 such that A7: for p being Point of X holds ( g2 . p = a & g2 is continuous ) by JGRAPH_2:20; consider g4 being Function of X,R^1 such that A8: for p being Point of X for r1, r0 being real number st g3 . p = r1 & g2 . p = r0 holds g4 . p = r1 - r0 and A9: g4 is continuous by A7, A5, JGRAPH_2:21; for q being Point of X holds g1 . q <> 0 by A2, A6; then consider g5 being Function of X,R^1 such that A10: for p being Point of X for r1, r0 being real number st g4 . p = r1 & g1 . p = r0 holds g5 . p = r1 / r0 and A11: g5 is continuous by A6, A9, JGRAPH_2:27; for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g5 . p = ((r1 / r2) - a) / b proof let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g5 . p = ((r1 / r2) - a) / b let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g5 . p = ((r1 / r2) - a) / b ) set r8 = r1 / r2; A12: g1 . p = b by A6; assume ( f1 . p = r1 & f2 . p = r2 ) ; ::_thesis: g5 . p = ((r1 / r2) - a) / b then A13: g3 . p = r1 / r2 by A4; g2 . p = a by A7; then g4 . p = (r1 / r2) - a by A8, A13; hence g5 . p = ((r1 / r2) - a) / b by A10, A12; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = ((r1 / r2) - a) / b ) & g is continuous ) by A11; ::_thesis: verum end; theorem Th5: :: JGRAPH_4:5 for X being non empty TopSpace for f1, f2 being Function of X,R^1 for a, b being Real st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (((r1 / r2) - a) / b) ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 for a, b being Real st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (((r1 / r2) - a) / b) ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: for a, b being Real st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (((r1 / r2) - a) / b) ) & g is continuous ) let a, b be Real; ::_thesis: ( f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (((r1 / r2) - a) / b) ) & g is continuous ) ) assume that A1: f1 is continuous and A2: f2 is continuous and A3: b <> 0 and A4: for q being Point of X holds f2 . q <> 0 ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (((r1 / r2) - a) / b) ) & g is continuous ) consider g3 being Function of X,R^1 such that A5: for p being Point of X for r1, r0 being real number st f1 . p = r1 & f2 . p = r0 holds g3 . p = r1 / r0 and A6: g3 is continuous by A1, A2, A4, JGRAPH_2:27; consider g1 being Function of X,R^1 such that A7: for p being Point of X holds ( g1 . p = b & g1 is continuous ) by JGRAPH_2:20; consider g2 being Function of X,R^1 such that A8: for p being Point of X holds ( g2 . p = a & g2 is continuous ) by JGRAPH_2:20; consider g4 being Function of X,R^1 such that A9: for p being Point of X for r1, r0 being real number st g3 . p = r1 & g2 . p = r0 holds g4 . p = r1 - r0 and A10: g4 is continuous by A8, A6, JGRAPH_2:21; for q being Point of X holds g1 . q <> 0 by A3, A7; then consider g5 being Function of X,R^1 such that A11: for p being Point of X for r1, r0 being real number st g4 . p = r1 & g1 . p = r0 holds g5 . p = r1 / r0 and A12: g5 is continuous by A7, A10, JGRAPH_2:27; consider g6 being Function of X,R^1 such that A13: for p being Point of X for r1, r0 being real number st f2 . p = r1 & g5 . p = r0 holds g6 . p = r1 * r0 and A14: g6 is continuous by A2, A12, JGRAPH_2:25; for p being Point of X for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds g6 . p = r2 * (((r1 / r2) - a) / b) proof let p be Point of X; ::_thesis: for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds g6 . p = r2 * (((r1 / r2) - a) / b) let r1, r2 be Real; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g6 . p = r2 * (((r1 / r2) - a) / b) ) assume that A15: f1 . p = r1 and A16: f2 . p = r2 ; ::_thesis: g6 . p = r2 * (((r1 / r2) - a) / b) A17: g2 . p = a by A8; set r8 = r1 / r2; A18: g1 . p = b by A7; g3 . p = r1 / r2 by A5, A15, A16; then g4 . p = (r1 / r2) - a by A9, A17; then g5 . p = ((r1 / r2) - a) / b by A11, A18; hence g6 . p = r2 * (((r1 / r2) - a) / b) by A13, A16; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (((r1 / r2) - a) / b) ) & g is continuous ) by A14; ::_thesis: verum end; theorem Th6: :: JGRAPH_4:6 for X being non empty TopSpace for f1 being Function of X,R^1 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 ^2 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 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 ^2 ) & g is continuous ) let f1 be Function of X,R^1; ::_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 ^2 ) & 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 ^2 ) & 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 * r1 and A2: g1 is continuous by JGRAPH_2:22; for p being Point of X for r1 being real number st f1 . p = r1 holds g1 . p = r1 ^2 by A1; 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 ^2 ) & g is continuous ) by A2; ::_thesis: verum end; theorem Th7: :: JGRAPH_4:7 for X being non empty TopSpace for f1 being Function of X,R^1 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 = abs r1 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 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 = abs r1 ) & g is continuous ) let f1 be Function of X,R^1; ::_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 = abs 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 = abs 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 ^2 and A2: g1 is continuous by Th6; for q being Point of X ex r being real number st ( g1 . q = r & r >= 0 ) proof let q be Point of X; ::_thesis: ex r being real number st ( g1 . q = r & r >= 0 ) reconsider r11 = f1 . q as Real by TOPMETR:17; g1 . q = r11 ^2 by A1; hence ex r being real number st ( g1 . q = r & r >= 0 ) by XREAL_1:63; ::_thesis: verum end; then 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 = sqrt r1 and A4: g2 is continuous by A2, JGRAPH_3:5; for p being Point of X for r1 being real number st f1 . p = r1 holds g2 . p = abs r1 proof let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds g2 . p = abs r1 let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g2 . p = abs r1 ) assume f1 . p = r1 ; ::_thesis: g2 . p = abs r1 then g1 . p = r1 ^2 by A1; then g2 . p = sqrt (r1 ^2) by A3 .= abs r1 by COMPLEX1:72 ; hence g2 . p = abs 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 = abs r1 ) & g is continuous ) by A4; ::_thesis: verum end; theorem Th8: :: JGRAPH_4:8 for X being non empty TopSpace for f1 being Function of X,R^1 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 ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 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 ) & g is continuous ) let f1 be Function of X,R^1; ::_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 ) & g is continuous ) ) consider g1 being Function of X,R^1 such that A1: for p being Point of X holds g1 . p = 0 and A2: g1 is continuous by JGRAPH_2:20; 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 ) & g is continuous ) then consider g2 being Function of X,R^1 such that A3: for p being Point of X for r1, r2 being real number st g1 . p = r1 & f1 . p = r2 holds g2 . p = r1 - r2 and A4: g2 is continuous by A2, JGRAPH_2:21; for p being Point of X for r1 being real number st f1 . p = r1 holds g2 . p = - r1 proof let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds g2 . p = - r1 let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g2 . p = - r1 ) assume A5: f1 . p = r1 ; ::_thesis: g2 . p = - r1 g1 . p = 0 by A1; then g2 . p = 0 - r1 by A3, A5; hence g2 . p = - 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 = - r1 ) & g is continuous ) by A4; ::_thesis: verum end; theorem Th9: :: JGRAPH_4:9 for X being non empty TopSpace for f1, f2 being Function of X,R^1 for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) ) & g is continuous ) let a, b be real number ; ::_thesis: ( f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) ) & g is continuous ) ) assume that A1: f1 is continuous and A2: f2 is continuous and A3: ( b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) ) ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) ) & g is continuous ) consider g1 being Function of X,R^1 such that A4: for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g1 . p = ((r1 / r2) - a) / b and A5: g1 is continuous by A1, A2, A3, Th4; consider g2 being Function of X,R^1 such that A6: for p being Point of X for s being real number st g1 . p = s holds g2 . p = s ^2 and A7: g2 is continuous by A5, Th6; consider g0 being Function of X,R^1 such that A8: for p being Point of X holds g0 . p = 1 and A9: g0 is continuous by JGRAPH_2:20; consider g3 being Function of X,R^1 such that A10: for p being Point of X for s, t being real number st g0 . p = s & g2 . p = t holds g3 . p = s - t and A11: g3 is continuous by A7, A9, JGRAPH_2:21; consider g4 being Function of X,R^1 such that A12: for p being Point of X for s being real number st g3 . p = s holds g4 . p = abs s and A13: g4 is continuous by A11, Th7; for q being Point of X ex r being real number st ( g4 . q = r & r >= 0 ) proof let q be Point of X; ::_thesis: ex r being real number st ( g4 . q = r & r >= 0 ) reconsider s = g3 . q as Real by TOPMETR:17; g4 . q = abs s by A12; hence ex r being real number st ( g4 . q = r & r >= 0 ) by COMPLEX1:46; ::_thesis: verum end; then consider g5 being Function of X,R^1 such that A14: for p being Point of X for s being real number st g4 . p = s holds g5 . p = sqrt s and A15: g5 is continuous by A13, JGRAPH_3:5; consider g6 being Function of X,R^1 such that A16: for p being Point of X for s being real number st g5 . p = s holds g6 . p = - s and A17: g6 is continuous by A15, Th8; consider g7 being Function of X,R^1 such that A18: for p being Point of X for r1, r0 being real number st f2 . p = r1 & g6 . p = r0 holds g7 . p = r1 * r0 and A19: g7 is continuous by A2, A17, JGRAPH_2:25; for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g7 . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) proof let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g7 . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g7 . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) ) assume that A20: f1 . p = r1 and A21: f2 . p = r2 ; ::_thesis: g7 . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) A22: g0 . p = 1 by A8; g1 . p = ((r1 / r2) - a) / b by A4, A20, A21; then g2 . p = (((r1 / r2) - a) / b) ^2 by A6; then g3 . p = 1 - ((((r1 / r2) - a) / b) ^2) by A10, A22; then g4 . p = abs (1 - ((((r1 / r2) - a) / b) ^2)) by A12; then g5 . p = sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))) by A14; then g6 . p = - (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) by A16; hence g7 . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) by A18, A21; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))))) ) & g is continuous ) by A19; ::_thesis: verum end; theorem Th10: :: JGRAPH_4:10 for X being non empty TopSpace for f1, f2 being Function of X,R^1 for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) ) & g is continuous ) proof let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) ) & g is continuous ) let f1, f2 be Function of X,R^1; ::_thesis: for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) ) & g is continuous ) let a, b be real number ; ::_thesis: ( f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) ) & g is continuous ) ) assume that A1: f1 is continuous and A2: f2 is continuous and A3: ( b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) ) ; ::_thesis: ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) ) & g is continuous ) consider g1 being Function of X,R^1 such that A4: for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g1 . p = ((r1 / r2) - a) / b and A5: g1 is continuous by A1, A2, A3, Th4; consider g2 being Function of X,R^1 such that A6: for p being Point of X for s being real number st g1 . p = s holds g2 . p = s ^2 and A7: g2 is continuous by A5, Th6; consider g0 being Function of X,R^1 such that A8: for p being Point of X holds g0 . p = 1 and A9: g0 is continuous by JGRAPH_2:20; consider g3 being Function of X,R^1 such that A10: for p being Point of X for s, t being real number st g0 . p = s & g2 . p = t holds g3 . p = s - t and A11: g3 is continuous by A7, A9, JGRAPH_2:21; consider g4 being Function of X,R^1 such that A12: for p being Point of X for s being real number st g3 . p = s holds g4 . p = abs s and A13: g4 is continuous by A11, Th7; for q being Point of X ex r being real number st ( g4 . q = r & r >= 0 ) proof let q be Point of X; ::_thesis: ex r being real number st ( g4 . q = r & r >= 0 ) reconsider s = g3 . q as Real by TOPMETR:17; g4 . q = abs s by A12; hence ex r being real number st ( g4 . q = r & r >= 0 ) by COMPLEX1:46; ::_thesis: verum end; then consider g5 being Function of X,R^1 such that A14: for p being Point of X for s being real number st g4 . p = s holds g5 . p = sqrt s and A15: g5 is continuous by A13, JGRAPH_3:5; consider g7 being Function of X,R^1 such that A16: for p being Point of X for r1, r0 being real number st f2 . p = r1 & g5 . p = r0 holds g7 . p = r1 * r0 and A17: g7 is continuous by A2, A15, JGRAPH_2:25; for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g7 . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) proof let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g7 . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g7 . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) ) assume that A18: f1 . p = r1 and A19: f2 . p = r2 ; ::_thesis: g7 . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) A20: g0 . p = 1 by A8; g1 . p = ((r1 / r2) - a) / b by A4, A18, A19; then g2 . p = (((r1 / r2) - a) / b) ^2 by A6; then g3 . p = 1 - ((((r1 / r2) - a) / b) ^2) by A10, A20; then g4 . p = abs (1 - ((((r1 / r2) - a) / b) ^2)) by A12; then g5 . p = sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2))) by A14; hence g7 . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) by A16, A19; ::_thesis: verum end; hence ex g being Function of X,R^1 st ( ( for p being Point of X for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r2 * (sqrt (abs (1 - ((((r1 / r2) - a) / b) ^2)))) ) & g is continuous ) by A17; ::_thesis: verum end; definition let n be Nat; deffunc H1( Point of (TOP-REAL n)) -> Element of REAL = |.$1.|; funcn NormF -> Function of (TOP-REAL n),R^1 means :Def1: :: JGRAPH_4:def 1 for q being Point of (TOP-REAL n) holds it . q = |.q.|; existence ex b1 being Function of (TOP-REAL n),R^1 st for q being Point of (TOP-REAL n) holds b1 . q = |.q.| proof A1: for x being Element of (TOP-REAL n) holds H1(x) in the carrier of R^1 by TOPMETR:17; thus ex IT being Function of (TOP-REAL n),R^1 st for q being Point of (TOP-REAL n) holds IT . q = H1(q) from FUNCT_2:sch_8(A1); ::_thesis: verum end; uniqueness for b1, b2 being Function of (TOP-REAL n),R^1 st ( for q being Point of (TOP-REAL n) holds b1 . q = |.q.| ) & ( for q being Point of (TOP-REAL n) holds b2 . q = |.q.| ) holds b1 = b2 proof thus for f, g being Function of (TOP-REAL n),R^1 st ( for q being Point of (TOP-REAL n) holds f . q = H1(q) ) & ( for q being Point of (TOP-REAL n) holds g . q = H1(q) ) holds f = g from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def1 defines NormF JGRAPH_4:def_1_:_ for n being Nat for b2 being Function of (TOP-REAL n),R^1 holds ( b2 = n NormF iff for q being Point of (TOP-REAL n) holds b2 . q = |.q.| ); theorem :: JGRAPH_4:11 for n being Nat holds ( dom (n NormF) = the carrier of (TOP-REAL n) & dom (n NormF) = REAL n ) proof let n be Nat; ::_thesis: ( dom (n NormF) = the carrier of (TOP-REAL n) & dom (n NormF) = REAL n ) thus dom (n NormF) = the carrier of (TOP-REAL n) by FUNCT_2:def_1; ::_thesis: dom (n NormF) = REAL n hence dom (n NormF) = REAL n by EUCLID:22; ::_thesis: verum end; theorem Th12: :: JGRAPH_4:12 for n being Nat holds n NormF is continuous proof let n be Nat; ::_thesis: n NormF is continuous A1: n in NAT by ORDINAL1:def_12; for q being Point of (TOP-REAL n) holds (n NormF) . q = |.q.| by Def1; hence n NormF is continuous by A1, JORDAN2C:83; ::_thesis: verum end; registration let n be Nat; clustern NormF -> continuous ; coherence n NormF is continuous by Th12; end; theorem Th13: :: JGRAPH_4:13 for n being Element of NAT for K0 being Subset of (TOP-REAL n) for f being Function of ((TOP-REAL n) | K0),R^1 st ( for p being Point of ((TOP-REAL n) | K0) holds f . p = (n NormF) . p ) holds f is continuous proof let n be Element of NAT ; ::_thesis: for K0 being Subset of (TOP-REAL n) for f being Function of ((TOP-REAL n) | K0),R^1 st ( for p being Point of ((TOP-REAL n) | K0) holds f . p = (n NormF) . p ) holds f is continuous let K0 be Subset of (TOP-REAL n); ::_thesis: for f being Function of ((TOP-REAL n) | K0),R^1 st ( for p being Point of ((TOP-REAL n) | K0) holds f . p = (n NormF) . p ) holds f is continuous let f be Function of ((TOP-REAL n) | K0),R^1; ::_thesis: ( ( for p being Point of ((TOP-REAL n) | K0) holds f . p = (n NormF) . p ) implies f is continuous ) A1: the carrier of (TOP-REAL n) /\ K0 = K0 by XBOOLE_1:28; reconsider g = n NormF as Function of (TOP-REAL n),R^1 ; assume for p being Point of ((TOP-REAL n) | K0) holds f . p = (n NormF) . p ; ::_thesis: f is continuous then A2: for x being set st x in dom f holds f . x = (n NormF) . x ; ( dom f = the carrier of ((TOP-REAL n) | K0) & the carrier of ((TOP-REAL n) | K0) = K0 ) by FUNCT_2:def_1, PRE_TOPC:8; then dom f = (dom (n NormF)) /\ K0 by A1, FUNCT_2:def_1; then f = g | K0 by A2, FUNCT_1:46; hence f is continuous by TOPMETR:7; ::_thesis: verum end; theorem Th14: :: JGRAPH_4:14 for n being Element of NAT for p being Point of (Euclid n) for r being Real for B being Subset of (TOP-REAL n) st B = cl_Ball (p,r) holds ( B is bounded & B is closed ) proof let n be Element of NAT ; ::_thesis: for p being Point of (Euclid n) for r being Real for B being Subset of (TOP-REAL n) st B = cl_Ball (p,r) holds ( B is bounded & B is closed ) let p be Point of (Euclid n); ::_thesis: for r being Real for B being Subset of (TOP-REAL n) st B = cl_Ball (p,r) holds ( B is bounded & B is closed ) let r be Real; ::_thesis: for B being Subset of (TOP-REAL n) st B = cl_Ball (p,r) holds ( B is bounded & B is closed ) let B be Subset of (TOP-REAL n); ::_thesis: ( B = cl_Ball (p,r) implies ( B is bounded & B is closed ) ) assume A1: B = cl_Ball (p,r) ; ::_thesis: ( B is bounded & B is closed ) cl_Ball (p,r) c= Ball (p,(r + 1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in cl_Ball (p,r) or x in Ball (p,(r + 1)) ) A2: r < r + 1 by XREAL_1:29; assume A3: x in cl_Ball (p,r) ; ::_thesis: x in Ball (p,(r + 1)) then reconsider q = x as Point of (Euclid n) ; dist (p,q) <= r by A3, METRIC_1:12; then dist (p,q) < r + 1 by A2, XXREAL_0:2; hence x in Ball (p,(r + 1)) by METRIC_1:11; ::_thesis: verum end; then cl_Ball (p,r) is bounded by TBSP_1:14; hence B is bounded by A1, JORDAN2C:11; ::_thesis: B is closed A4: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; then reconsider BB = B as Subset of (TopSpaceMetr (Euclid n)) ; BB is closed by A1, TOPREAL6:57; hence B is closed by A4, PRE_TOPC:31; ::_thesis: verum end; theorem Th15: :: JGRAPH_4:15 for p being Point of (Euclid 2) for r being Real for B being Subset of (TOP-REAL 2) st B = cl_Ball (p,r) holds B is compact proof let p be Point of (Euclid 2); ::_thesis: for r being Real for B being Subset of (TOP-REAL 2) st B = cl_Ball (p,r) holds B is compact let r be Real; ::_thesis: for B being Subset of (TOP-REAL 2) st B = cl_Ball (p,r) holds B is compact let B be Subset of (TOP-REAL 2); ::_thesis: ( B = cl_Ball (p,r) implies B is compact ) assume B = cl_Ball (p,r) ; ::_thesis: B is compact then ( B is bounded & B is closed ) by Th14; hence B is compact by TOPREAL6:79; ::_thesis: verum end; begin definition let s be real number ; let q be Point of (TOP-REAL 2); func FanW (s,q) -> Point of (TOP-REAL 2) equals :Def2: :: JGRAPH_4:def 2 |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2)))),((((q `2) / |.q.|) - s) / (1 - s))]| if ( (q `2) / |.q.| >= s & q `1 < 0 ) |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2)))),((((q `2) / |.q.|) - s) / (1 + s))]| if ( (q `2) / |.q.| < s & q `1 < 0 ) otherwise q; correctness coherence ( ( (q `2) / |.q.| >= s & q `1 < 0 implies |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2)))),((((q `2) / |.q.|) - s) / (1 - s))]| is Point of (TOP-REAL 2) ) & ( (q `2) / |.q.| < s & q `1 < 0 implies |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2)))),((((q `2) / |.q.|) - s) / (1 + s))]| is Point of (TOP-REAL 2) ) & ( ( not (q `2) / |.q.| >= s or not q `1 < 0 ) & ( not (q `2) / |.q.| < s or not q `1 < 0 ) implies q is Point of (TOP-REAL 2) ) ); consistency for b1 being Point of (TOP-REAL 2) st (q `2) / |.q.| >= s & q `1 < 0 & (q `2) / |.q.| < s & q `1 < 0 holds ( b1 = |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2)))),((((q `2) / |.q.|) - s) / (1 - s))]| iff b1 = |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2)))),((((q `2) / |.q.|) - s) / (1 + s))]| ); ; end; :: deftheorem Def2 defines FanW JGRAPH_4:def_2_:_ for s being real number for q being Point of (TOP-REAL 2) holds ( ( (q `2) / |.q.| >= s & q `1 < 0 implies FanW (s,q) = |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2)))),((((q `2) / |.q.|) - s) / (1 - s))]| ) & ( (q `2) / |.q.| < s & q `1 < 0 implies FanW (s,q) = |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2)))),((((q `2) / |.q.|) - s) / (1 + s))]| ) & ( ( not (q `2) / |.q.| >= s or not q `1 < 0 ) & ( not (q `2) / |.q.| < s or not q `1 < 0 ) implies FanW (s,q) = q ) ); definition let s be real number ; funcs -FanMorphW -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def3: :: JGRAPH_4:def 3 for q being Point of (TOP-REAL 2) holds it . q = FanW (s,q); existence ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st for q being Point of (TOP-REAL 2) holds b1 . q = FanW (s,q) proof deffunc H1( Point of (TOP-REAL 2)) -> Point of (TOP-REAL 2) = FanW (s,$1); thus ex IT being Function of (TOP-REAL 2),(TOP-REAL 2) st for q being Point of (TOP-REAL 2) holds IT . q = H1(q) from FUNCT_2:sch_4(); ::_thesis: verum end; uniqueness for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for q being Point of (TOP-REAL 2) holds b1 . q = FanW (s,q) ) & ( for q being Point of (TOP-REAL 2) holds b2 . q = FanW (s,q) ) holds b1 = b2 proof deffunc H1( Point of (TOP-REAL 2)) -> Point of (TOP-REAL 2) = FanW (s,$1); thus for f, g being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for q being Point of (TOP-REAL 2) holds f . q = H1(q) ) & ( for q being Point of (TOP-REAL 2) holds g . q = H1(q) ) holds f = g from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def3 defines -FanMorphW JGRAPH_4:def_3_:_ for s being real number for b2 being Function of (TOP-REAL 2),(TOP-REAL 2) holds ( b2 = s -FanMorphW iff for q being Point of (TOP-REAL 2) holds b2 . q = FanW (s,q) ); theorem Th16: :: JGRAPH_4:16 for q being Point of (TOP-REAL 2) for sn being real number holds ( ( (q `2) / |.q.| >= sn & q `1 < 0 implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 >= 0 implies (sn -FanMorphW) . q = q ) ) proof let q be Point of (TOP-REAL 2); ::_thesis: for sn being real number holds ( ( (q `2) / |.q.| >= sn & q `1 < 0 implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 >= 0 implies (sn -FanMorphW) . q = q ) ) let sn be real number ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 < 0 implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 >= 0 implies (sn -FanMorphW) . q = q ) ) hereby ::_thesis: ( q `1 >= 0 implies (sn -FanMorphW) . q = q ) assume ( (q `2) / |.q.| >= sn & q `1 < 0 ) ; ::_thesis: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| then FanW (sn,q) = |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),((((q `2) / |.q.|) - sn) / (1 - sn))]| by Def2 .= |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by EUCLID:58 ; hence (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by Def3; ::_thesis: verum end; assume A1: q `1 >= 0 ; ::_thesis: (sn -FanMorphW) . q = q (sn -FanMorphW) . q = FanW (sn,q) by Def3; hence (sn -FanMorphW) . q = q by A1, Def2; ::_thesis: verum end; theorem Th17: :: JGRAPH_4:17 for q being Point of (TOP-REAL 2) for sn being Real st (q `2) / |.q.| <= sn & q `1 < 0 holds (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| proof let q be Point of (TOP-REAL 2); ::_thesis: for sn being Real st (q `2) / |.q.| <= sn & q `1 < 0 holds (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| let sn be Real; ::_thesis: ( (q `2) / |.q.| <= sn & q `1 < 0 implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) assume that A1: (q `2) / |.q.| <= sn and A2: q `1 < 0 ; ::_thesis: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| percases ( (q `2) / |.q.| < sn or (q `2) / |.q.| = sn ) by A1, XXREAL_0:1; suppose (q `2) / |.q.| < sn ; ::_thesis: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| then FanW (sn,q) = |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),((((q `2) / |.q.|) - sn) / (1 + sn))]| by A2, Def2 .= |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by EUCLID:58 ; hence (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by Def3; ::_thesis: verum end; supposeA3: (q `2) / |.q.| = sn ; ::_thesis: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| then (((q `2) / |.q.|) - sn) / (1 - sn) = 0 ; hence (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A2, A3, Th16; ::_thesis: verum end; end; end; theorem Th18: :: JGRAPH_4:18 for q being Point of (TOP-REAL 2) for sn being Real st - 1 < sn & sn < 1 holds ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) proof let q be Point of (TOP-REAL 2); ::_thesis: for sn being Real st - 1 < sn & sn < 1 holds ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) ) assume that A1: - 1 < sn and A2: sn < 1 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) percases ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) ) or ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) ) or q `1 > 0 or q = 0. (TOP-REAL 2) ) ; supposeA3: ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) percases ( q `1 < 0 or q `1 >= 0 ) ; supposeA4: q `1 < 0 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) then FanW (sn,q) = |.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),((((q `2) / |.q.|) - sn) / (1 - sn))]| by A3, Def2 .= |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by EUCLID:58 ; hence ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) by A4, Def3, Th17; ::_thesis: verum end; supposeA5: q `1 >= 0 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) then A6: (sn -FanMorphW) . q = q by Th16; A7: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; A8: 1 - sn > 0 by A2, XREAL_1:149; A9: q `1 = 0 by A3, A5; |.q.| <> 0 by A3, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `2) ^2) / (|.q.| ^2) = 1 ^2 by A7, A9, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 = 1 ^2 by XCMPLX_1:76; then A10: sqrt (((q `2) / |.q.|) ^2) = 1 by SQUARE_1:22; A11: now__::_thesis:_not_q_`2_<_0 assume q `2 < 0 ; ::_thesis: contradiction then - ((q `2) / |.q.|) = 1 by A10, SQUARE_1:23; hence contradiction by A1, A3; ::_thesis: verum end; sqrt (|.q.| ^2) = |.q.| by SQUARE_1:22; then A12: |.q.| = q `2 by A7, A9, A11, SQUARE_1:22; then 1 = (q `2) / |.q.| by A3, TOPRNS_1:24, XCMPLX_1:60; then (((q `2) / |.q.|) - sn) / (1 - sn) = 1 by A8, XCMPLX_1:60; hence ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) by A2, A6, A9, A12, EUCLID:53, SQUARE_1:17, TOPRNS_1:24, XCMPLX_1:60; ::_thesis: verum end; end; end; supposeA13: ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) percases ( q `1 < 0 or q `1 >= 0 ) ; suppose q `1 < 0 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) hence ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) by Th16, Th17; ::_thesis: verum end; supposeA14: q `1 >= 0 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) A15: 1 + sn > 0 by A1, XREAL_1:148; A16: |.q.| <> 0 by A13, TOPRNS_1:24; A17: q `1 = 0 by A13, A14; ( |.q.| > 0 & 1 > (q `2) / |.q.| ) by A2, A13, Lm1, XXREAL_0:2; then 1 * |.q.| > ((q `2) / |.q.|) * |.q.| by XREAL_1:68; then A18: ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & |.q.| > q `2 ) by A13, JGRAPH_3:1, TOPRNS_1:24, XCMPLX_1:87; then A19: q `2 = - |.q.| by A17, SQUARE_1:40; then - 1 = (q `2) / |.q.| by A13, TOPRNS_1:24, XCMPLX_1:197; then A20: (((q `2) / |.q.|) - sn) / (1 + sn) = (- (1 + sn)) / (1 + sn) .= - 1 by A15, XCMPLX_1:197 ; |.q.| = - (q `2) by A17, A18, SQUARE_1:40; then |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| = q by A17, A20, EUCLID:53, SQUARE_1:17; hence ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) by A1, A14, A16, A19, Th16, XCMPLX_1:197; ::_thesis: verum end; end; end; suppose ( q `1 > 0 or q = 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) hence ( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) ; ::_thesis: verum end; end; end; Lm2: for K being non empty Subset of (TOP-REAL 2) holds ( proj1 | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds (proj1 | K) . q = proj1 . q ) ) proof let K be non empty Subset of (TOP-REAL 2); ::_thesis: ( proj1 | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds (proj1 | K) . q = proj1 . q ) ) reconsider g2 = proj1 | K as Function of ((TOP-REAL 2) | K),R^1 by TOPMETR:17; A1: the carrier of ((TOP-REAL 2) | K) = K by PRE_TOPC:8; for q being Point of ((TOP-REAL 2) | K) holds g2 . q = proj1 . q proof let q be Point of ((TOP-REAL 2) | K); ::_thesis: g2 . q = proj1 . q ( q in the carrier of ((TOP-REAL 2) | K) & dom proj1 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1; then q in (dom proj1) /\ K by A1, XBOOLE_0:def_4; hence g2 . q = proj1 . q by FUNCT_1:48; ::_thesis: verum end; hence ( proj1 | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds (proj1 | K) . q = proj1 . q ) ) by JGRAPH_2:29; ::_thesis: verum end; Lm3: for K being non empty Subset of (TOP-REAL 2) holds ( proj2 | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds (proj2 | K) . q = proj2 . q ) ) proof let K be non empty Subset of (TOP-REAL 2); ::_thesis: ( proj2 | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds (proj2 | K) . q = proj2 . q ) ) reconsider g2 = proj2 | K as Function of ((TOP-REAL 2) | K),R^1 by TOPMETR:17; A1: the carrier of ((TOP-REAL 2) | K) = K by PRE_TOPC:8; for q being Point of ((TOP-REAL 2) | K) holds g2 . q = proj2 . q proof let q be Point of ((TOP-REAL 2) | K); ::_thesis: g2 . q = proj2 . q ( q in the carrier of ((TOP-REAL 2) | K) & dom proj2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1; then q in (dom proj2) /\ K by A1, XBOOLE_0:def_4; hence g2 . q = proj2 . q by FUNCT_1:48; ::_thesis: verum end; hence ( proj2 | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds (proj2 | K) . q = proj2 . q ) ) by JGRAPH_2:30; ::_thesis: verum end; Lm4: dom (2 NormF) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; Lm5: for K being non empty Subset of (TOP-REAL 2) holds ( (2 NormF) | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds ((2 NormF) | K) . q = (2 NormF) . q ) ) proof let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: ( (2 NormF) | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1 & ( for q being Point of ((TOP-REAL 2) | K1) holds ((2 NormF) | K1) . q = (2 NormF) . q ) ) A1: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then reconsider g1 = (2 NormF) | K1 as Function of ((TOP-REAL 2) | K1),R^1 by FUNCT_2:32; for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = (2 NormF) . q proof let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = (2 NormF) . q q in the carrier of ((TOP-REAL 2) | K1) ; then q in (dom (2 NormF)) /\ K1 by A1, Lm4, XBOOLE_0:def_4; hence g1 . q = (2 NormF) . q by FUNCT_1:48; ::_thesis: verum end; hence ( (2 NormF) | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1 & ( for q being Point of ((TOP-REAL 2) | K1) holds ((2 NormF) | K1) . q = (2 NormF) . q ) ) by Th13; ::_thesis: verum end; Lm6: for K1 being non empty Subset of (TOP-REAL 2) for g1 being Function of ((TOP-REAL 2) | K1),R^1 st g1 = (2 NormF) | K1 & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) ) holds for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 proof let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for g1 being Function of ((TOP-REAL 2) | K1),R^1 st g1 = (2 NormF) | K1 & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) ) holds for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 let g1 be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( g1 = (2 NormF) | K1 & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) ) implies for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 ) assume that A1: g1 = (2 NormF) | K1 and A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) ; ::_thesis: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0 ( the carrier of ((TOP-REAL 2) | K1) = K1 & q in the carrier of ((TOP-REAL 2) | K1) ) by PRE_TOPC:8; then reconsider q2 = q as Point of (TOP-REAL 2) ; g1 . q = (2 NormF) . q by A1, Lm5 .= |.q2.| by Def1 ; hence g1 . q <> 0 by A2, TOPRNS_1:24; ::_thesis: verum end; theorem Th19: :: JGRAPH_4:19 for sn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let sn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = sn; set b = 1 - sn; reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm3; assume that A1: sn < 1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 - sn > 0 by A1, XREAL_1:149; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (((r1 / r2) - sn) / (1 - sn)) and A6: g3 is continuous by A4, Th5; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in K1 by A7, A8, A9, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( proj2 . r = r `2 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_6; A11: ( g2 . s = proj2 . s & g1 . s = (2 NormF) . s ) by Lm3, Lm5; f . r = |.r.| * ((((r `2) / |.r.|) - sn) / (1 - sn)) by A2, A9; hence f . x = g3 . x by A5, A11, A10; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th20: :: JGRAPH_4:20 for sn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let sn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = sn; set b = 1 + sn; reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm3; assume that A1: - 1 < sn and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 + sn > 0 by A1, XREAL_1:148; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (((r1 / r2) - sn) / (1 + sn)) and A6: g3 is continuous by A4, Th5; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; A8: for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in dom g3 by A7, A9; then x in K1 by A7, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( proj2 . r = r `2 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_6; A11: ( g2 . s = proj2 . s & g1 . s = (2 NormF) . s ) by Lm3, Lm5; f . r = |.r.| * ((((r `2) / |.r.|) - sn) / (1 + sn)) by A2, A9; hence f . x = g3 . x by A5, A11, A10; ::_thesis: verum end; dom f = dom g3 by A7, FUNCT_2:def_1; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th21: :: JGRAPH_4:21 for sn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let sn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = sn; set b = 1 - sn; reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm3; assume that A1: sn < 1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 - sn > 0 by A1, XREAL_1:149; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - sn) / (1 - sn)) ^2))))) and A6: g3 is continuous by A4, Th9; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) A9: 1 - sn > 0 by A1, XREAL_1:149; assume A10: x in dom f ; ::_thesis: f . x = g3 . x then x in K1 by A7, A8, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A11: |.r.| <> 0 by A3, A10, TOPRNS_1:24; |.r.| ^2 = ((r `1) ^2) + ((r `2) ^2) by JGRAPH_3:1; then A12: ((r `2) - |.r.|) * ((r `2) + |.r.|) = - ((r `1) ^2) ; (r `1) ^2 >= 0 by XREAL_1:63; then r `2 <= |.r.| by A12, XREAL_1:93; then (r `2) / |.r.| <= |.r.| / |.r.| by XREAL_1:72; then (r `2) / |.r.| <= 1 by A11, XCMPLX_1:60; then A13: ((r `2) / |.r.|) - sn <= 1 - sn by XREAL_1:9; reconsider s = x as Point of ((TOP-REAL 2) | K1) by A10; A14: now__::_thesis:_not_(1_-_sn)_^2_=_0 assume (1 - sn) ^2 = 0 ; ::_thesis: contradiction then (1 - sn) + sn = 0 + sn by XCMPLX_1:6; hence contradiction by A1; ::_thesis: verum end; sn - ((r `2) / |.r.|) <= 0 by A3, A10, XREAL_1:47; then - (sn - ((r `2) / |.r.|)) >= - (1 - sn) by A9, XREAL_1:24; then ( (1 - sn) ^2 >= 0 & (((r `2) / |.r.|) - sn) ^2 <= (1 - sn) ^2 ) by A13, SQUARE_1:49, XREAL_1:63; then ((((r `2) / |.r.|) - sn) ^2) / ((1 - sn) ^2) <= ((1 - sn) ^2) / ((1 - sn) ^2) by XREAL_1:72; then ((((r `2) / |.r.|) - sn) ^2) / ((1 - sn) ^2) <= 1 by A14, XCMPLX_1:60; then ((((r `2) / |.r.|) - sn) / (1 - sn)) ^2 <= 1 by XCMPLX_1:76; then 1 - (((((r `2) / |.r.|) - sn) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then abs (1 - (((((r `2) / |.r.|) - sn) / (1 - sn)) ^2)) = 1 - (((((r `2) / |.r.|) - sn) / (1 - sn)) ^2) by ABSVALUE:def_1; then A15: f . r = |.r.| * (- (sqrt (abs (1 - (((((r `2) / |.r.|) - sn) / (1 - sn)) ^2))))) by A2, A10; A16: ( proj2 . r = r `2 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_6; ( g2 . s = proj2 . s & g1 . s = (2 NormF) . s ) by Lm3, Lm5; hence f . x = g3 . x by A5, A15, A16; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th22: :: JGRAPH_4:22 for sn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let sn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = sn; set b = 1 + sn; reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm3; assume that A1: - 1 < sn and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous A4: 1 + sn > 0 by A1, XREAL_1:148; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - sn) / (1 + sn)) ^2))))) and A6: g3 is continuous by A4, Th9; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then x in K1 by A7, A8, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; reconsider s = x as Point of ((TOP-REAL 2) | K1) by A9; A10: (1 + sn) ^2 > 0 by A4, SQUARE_1:12; A11: |.r.| <> 0 by A3, A9, TOPRNS_1:24; |.r.| ^2 = ((r `1) ^2) + ((r `2) ^2) by JGRAPH_3:1; then A12: ((r `2) - |.r.|) * ((r `2) + |.r.|) = - ((r `1) ^2) ; (r `1) ^2 >= 0 by XREAL_1:63; then - |.r.| <= r `2 by A12, XREAL_1:93; then (r `2) / |.r.| >= (- |.r.|) / |.r.| by XREAL_1:72; then (r `2) / |.r.| >= - 1 by A11, XCMPLX_1:197; then ((r `2) / |.r.|) - sn >= (- 1) - sn by XREAL_1:9; then A13: ((r `2) / |.r.|) - sn >= - (1 + sn) ; sn - ((r `2) / |.r.|) >= 0 by A3, A9, XREAL_1:48; then - (sn - ((r `2) / |.r.|)) <= - 0 ; then (((r `2) / |.r.|) - sn) ^2 <= (1 + sn) ^2 by A4, A13, SQUARE_1:49; then ((((r `2) / |.r.|) - sn) ^2) / ((1 + sn) ^2) <= ((1 + sn) ^2) / ((1 + sn) ^2) by A4, XREAL_1:72; then ((((r `2) / |.r.|) - sn) ^2) / ((1 + sn) ^2) <= 1 by A10, XCMPLX_1:60; then ((((r `2) / |.r.|) - sn) / (1 + sn)) ^2 <= 1 by XCMPLX_1:76; then 1 - (((((r `2) / |.r.|) - sn) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then abs (1 - (((((r `2) / |.r.|) - sn) / (1 + sn)) ^2)) = 1 - (((((r `2) / |.r.|) - sn) / (1 + sn)) ^2) by ABSVALUE:def_1; then A14: f . r = |.r.| * (- (sqrt (abs (1 - (((((r `2) / |.r.|) - sn) / (1 + sn)) ^2))))) by A2, A9; A15: ( proj2 . r = r `2 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_6; ( g2 . s = proj2 . s & g1 . s = (2 NormF) . s ) by Lm3, Lm5; hence f . x = g3 . x by A5, A14, A15; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th23: :: JGRAPH_4:23 for sn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set cn = sqrt (1 - (sn ^2)); set p0 = |[(- (sqrt (1 - (sn ^2)))),sn]|; A1: |[(- (sqrt (1 - (sn ^2)))),sn]| `1 = - (sqrt (1 - (sn ^2))) by EUCLID:52; |[(- (sqrt (1 - (sn ^2)))),sn]| `2 = sn by EUCLID:52; then A2: |.|[(- (sqrt (1 - (sn ^2)))),sn]|.| = sqrt (((- (sqrt (1 - (sn ^2)))) ^2) + (sn ^2)) by A1, JGRAPH_3:1 .= sqrt (((sqrt (1 - (sn ^2))) ^2) + (sn ^2)) ; assume A3: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then sn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (sn ^2) > 0 by XREAL_1:50; then A5: - (- (sqrt (1 - (sn ^2)))) > 0 by SQUARE_1:25; A6: now__::_thesis:_not_|[(-_(sqrt_(1_-_(sn_^2)))),sn]|_=_0._(TOP-REAL_2) assume |[(- (sqrt (1 - (sn ^2)))),sn]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then - (- (sqrt (1 - (sn ^2)))) = - 0 by EUCLID:52, JGRAPH_2:3; hence contradiction by A4, SQUARE_1:25; ::_thesis: verum end; (sqrt (1 - (sn ^2))) ^2 = 1 - (sn ^2) by A4, SQUARE_1:def_2; then (|[(- (sqrt (1 - (sn ^2)))),sn]| `2) / |.|[(- (sqrt (1 - (sn ^2)))),sn]|.| = sn by A2, EUCLID:52, SQUARE_1:18; then A7: |[(- (sqrt (1 - (sn ^2)))),sn]| in K0 by A3, A1, A6, A5; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A8: rng (proj1 * ((sn -FanMorphW) | K1)) c= the carrier of R^1 by TOPMETR:17; A9: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & (p8 `2) / |.p8.| >= sn & p8 `1 <= 0 & p8 <> 0. (TOP-REAL 2) ) by A3; hence x in B0 by A3; ::_thesis: verum end; A10: dom ((sn -FanMorphW) | K1) c= dom (proj2 * ((sn -FanMorphW) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((sn -FanMorphW) | K1) or x in dom (proj2 * ((sn -FanMorphW) | K1)) ) assume A11: x in dom ((sn -FanMorphW) | K1) ; ::_thesis: x in dom (proj2 * ((sn -FanMorphW) | K1)) then x in (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61; then x in dom (sn -FanMorphW) by XBOOLE_0:def_4; then A12: ( dom proj2 = the carrier of (TOP-REAL 2) & (sn -FanMorphW) . x in rng (sn -FanMorphW) ) by FUNCT_1:3, FUNCT_2:def_1; ((sn -FanMorphW) | K1) . x = (sn -FanMorphW) . x by A11, FUNCT_1:47; hence x in dom (proj2 * ((sn -FanMorphW) | K1)) by A11, A12, FUNCT_1:11; ::_thesis: verum end; A13: rng (proj2 * ((sn -FanMorphW) | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj2 * ((sn -FanMorphW) | K1)) c= dom ((sn -FanMorphW) | K1) by RELAT_1:25; then dom (proj2 * ((sn -FanMorphW) | K1)) = dom ((sn -FanMorphW) | K1) by A10, XBOOLE_0:def_10 .= (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g2 = proj2 * ((sn -FanMorphW) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A13, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) A14: dom ((sn -FanMorphW) | K1) = (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A15: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A16: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A15; then A17: (sn -FanMorphW) . p = |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A3, Th18; ((sn -FanMorphW) | K1) . p = (sn -FanMorphW) . p by A16, A15, FUNCT_1:49; then g2 . p = proj2 . |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A16, A14, A15, A17, FUNCT_1:13 .= |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| `2 by PSCOMP_1:def_6 .= |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) by EUCLID:52 ; hence g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A18: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ; A19: dom ((sn -FanMorphW) | K1) c= dom (proj1 * ((sn -FanMorphW) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((sn -FanMorphW) | K1) or x in dom (proj1 * ((sn -FanMorphW) | K1)) ) assume A20: x in dom ((sn -FanMorphW) | K1) ; ::_thesis: x in dom (proj1 * ((sn -FanMorphW) | K1)) then x in (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61; then x in dom (sn -FanMorphW) by XBOOLE_0:def_4; then A21: ( dom proj1 = the carrier of (TOP-REAL 2) & (sn -FanMorphW) . x in rng (sn -FanMorphW) ) by FUNCT_1:3, FUNCT_2:def_1; ((sn -FanMorphW) | K1) . x = (sn -FanMorphW) . x by A20, FUNCT_1:47; hence x in dom (proj1 * ((sn -FanMorphW) | K1)) by A20, A21, FUNCT_1:11; ::_thesis: verum end; dom (proj1 * ((sn -FanMorphW) | K1)) c= dom ((sn -FanMorphW) | K1) by RELAT_1:25; then dom (proj1 * ((sn -FanMorphW) | K1)) = dom ((sn -FanMorphW) | K1) by A19, XBOOLE_0:def_10 .= (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g1 = proj1 * ((sn -FanMorphW) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A8, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) ) A22: dom ((sn -FanMorphW) | K1) = (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A23: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A24: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A23; then A25: (sn -FanMorphW) . p = |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A3, Th18; ((sn -FanMorphW) | K1) . p = (sn -FanMorphW) . p by A24, A23, FUNCT_1:49; then g1 . p = proj1 . |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A24, A22, A23, A25, FUNCT_1:13 .= |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| `1 by PSCOMP_1:def_5 .= |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52 ; hence g1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A26: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) ; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) A27: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A27; hence ( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then A28: f1 is continuous by A3, A26, Th21; A29: for x, y, r, s being real number st |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| proof let x, y, r, s be real number ; ::_thesis: ( |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| implies f . |[x,y]| = |[r,s]| ) assume that A30: |[x,y]| in K1 and A31: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[r,s]| set p99 = |[x,y]|; A32: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A30; A33: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then A34: f1 . |[x,y]| = |.|[x,y]|.| * (- (sqrt (1 - (((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 - sn)) ^2)))) by A26, A30; ((sn -FanMorphW) | K0) . |[x,y]| = (sn -FanMorphW) . |[x,y]| by A30, FUNCT_1:49 .= |[(|.|[x,y]|.| * (- (sqrt (1 - (((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 - sn)) ^2))))),(|.|[x,y]|.| * ((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 - sn)))]| by A3, A32, Th18 .= |[r,s]| by A18, A30, A31, A33, A34 ; hence f . |[x,y]| = |[r,s]| by A3; ::_thesis: verum end; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) A35: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A35; hence ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then f2 is continuous by A3, A18, Th19; hence f is continuous by A7, A9, A28, A29, JGRAPH_2:35; ::_thesis: verum end; theorem Th24: :: JGRAPH_4:24 for sn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set cn = sqrt (1 - (sn ^2)); set p0 = |[(- (sqrt (1 - (sn ^2)))),sn]|; A1: |[(- (sqrt (1 - (sn ^2)))),sn]| `1 = - (sqrt (1 - (sn ^2))) by EUCLID:52; |[(- (sqrt (1 - (sn ^2)))),sn]| `2 = sn by EUCLID:52; then A2: |.|[(- (sqrt (1 - (sn ^2)))),sn]|.| = sqrt (((- (sqrt (1 - (sn ^2)))) ^2) + (sn ^2)) by A1, JGRAPH_3:1 .= sqrt (((sqrt (1 - (sn ^2))) ^2) + (sn ^2)) ; assume A3: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then sn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (sn ^2) > 0 by XREAL_1:50; then A5: - (- (sqrt (1 - (sn ^2)))) > 0 by SQUARE_1:25; A6: now__::_thesis:_not_|[(-_(sqrt_(1_-_(sn_^2)))),sn]|_=_0._(TOP-REAL_2) assume |[(- (sqrt (1 - (sn ^2)))),sn]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then - (- (sqrt (1 - (sn ^2)))) = - 0 by EUCLID:52, JGRAPH_2:3; hence contradiction by A4, SQUARE_1:25; ::_thesis: verum end; (sqrt (1 - (sn ^2))) ^2 = 1 - (sn ^2) by A4, SQUARE_1:def_2; then (|[(- (sqrt (1 - (sn ^2)))),sn]| `2) / |.|[(- (sqrt (1 - (sn ^2)))),sn]|.| = sn by A2, EUCLID:52, SQUARE_1:18; then A7: |[(- (sqrt (1 - (sn ^2)))),sn]| in K0 by A3, A1, A6, A5; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A8: rng (proj1 * ((sn -FanMorphW) | K1)) c= the carrier of R^1 by TOPMETR:17; A9: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & (p8 `2) / |.p8.| <= sn & p8 `1 <= 0 & p8 <> 0. (TOP-REAL 2) ) by A3; hence x in B0 by A3; ::_thesis: verum end; A10: dom ((sn -FanMorphW) | K1) c= dom (proj2 * ((sn -FanMorphW) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((sn -FanMorphW) | K1) or x in dom (proj2 * ((sn -FanMorphW) | K1)) ) assume A11: x in dom ((sn -FanMorphW) | K1) ; ::_thesis: x in dom (proj2 * ((sn -FanMorphW) | K1)) then x in (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61; then x in dom (sn -FanMorphW) by XBOOLE_0:def_4; then A12: ( dom proj2 = the carrier of (TOP-REAL 2) & (sn -FanMorphW) . x in rng (sn -FanMorphW) ) by FUNCT_1:3, FUNCT_2:def_1; ((sn -FanMorphW) | K1) . x = (sn -FanMorphW) . x by A11, FUNCT_1:47; hence x in dom (proj2 * ((sn -FanMorphW) | K1)) by A11, A12, FUNCT_1:11; ::_thesis: verum end; A13: rng (proj2 * ((sn -FanMorphW) | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj2 * ((sn -FanMorphW) | K1)) c= dom ((sn -FanMorphW) | K1) by RELAT_1:25; then dom (proj2 * ((sn -FanMorphW) | K1)) = dom ((sn -FanMorphW) | K1) by A10, XBOOLE_0:def_10 .= (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g2 = proj2 * ((sn -FanMorphW) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A13, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) A14: dom ((sn -FanMorphW) | K1) = (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A15: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A16: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A15; then A17: (sn -FanMorphW) . p = |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A3, Th18; ((sn -FanMorphW) | K1) . p = (sn -FanMorphW) . p by A16, A15, FUNCT_1:49; then g2 . p = proj2 . |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A16, A14, A15, A17, FUNCT_1:13 .= |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| `2 by PSCOMP_1:def_6 .= |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) by EUCLID:52 ; hence g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A18: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ; A19: dom ((sn -FanMorphW) | K1) c= dom (proj1 * ((sn -FanMorphW) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((sn -FanMorphW) | K1) or x in dom (proj1 * ((sn -FanMorphW) | K1)) ) assume A20: x in dom ((sn -FanMorphW) | K1) ; ::_thesis: x in dom (proj1 * ((sn -FanMorphW) | K1)) then x in (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61; then x in dom (sn -FanMorphW) by XBOOLE_0:def_4; then A21: ( dom proj1 = the carrier of (TOP-REAL 2) & (sn -FanMorphW) . x in rng (sn -FanMorphW) ) by FUNCT_1:3, FUNCT_2:def_1; ((sn -FanMorphW) | K1) . x = (sn -FanMorphW) . x by A20, FUNCT_1:47; hence x in dom (proj1 * ((sn -FanMorphW) | K1)) by A20, A21, FUNCT_1:11; ::_thesis: verum end; dom (proj1 * ((sn -FanMorphW) | K1)) c= dom ((sn -FanMorphW) | K1) by RELAT_1:25; then dom (proj1 * ((sn -FanMorphW) | K1)) = dom ((sn -FanMorphW) | K1) by A19, XBOOLE_0:def_10 .= (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g1 = proj1 * ((sn -FanMorphW) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A8, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) ) A22: dom ((sn -FanMorphW) | K1) = (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A23: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A24: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A23; then A25: (sn -FanMorphW) . p = |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A3, Th18; ((sn -FanMorphW) | K1) . p = (sn -FanMorphW) . p by A24, A23, FUNCT_1:49; then g1 . p = proj1 . |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A24, A22, A23, A25, FUNCT_1:13 .= |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| `1 by PSCOMP_1:def_5 .= |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) by EUCLID:52 ; hence g1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A26: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) ; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) A27: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A27; hence ( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then A28: f1 is continuous by A3, A26, Th22; A29: for x, y, r, s being real number st |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| proof let x, y, r, s be real number ; ::_thesis: ( |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| implies f . |[x,y]| = |[r,s]| ) assume that A30: |[x,y]| in K1 and A31: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[r,s]| set p99 = |[x,y]|; A32: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A30; A33: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then A34: f1 . |[x,y]| = |.|[x,y]|.| * (- (sqrt (1 - (((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 + sn)) ^2)))) by A26, A30; ((sn -FanMorphW) | K0) . |[x,y]| = (sn -FanMorphW) . |[x,y]| by A30, FUNCT_1:49 .= |[(|.|[x,y]|.| * (- (sqrt (1 - (((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 + sn)) ^2))))),(|.|[x,y]|.| * ((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 + sn)))]| by A3, A32, Th18 .= |[r,s]| by A18, A30, A31, A33, A34 ; hence f . |[x,y]| = |[r,s]| by A3; ::_thesis: verum end; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) A35: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A35; hence ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then f2 is continuous by A3, A18, Th20; hence f is continuous by A7, A9, A28, A29, JGRAPH_2:35; ::_thesis: verum end; Lm7: for sn being Real for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 >= sn * |.p7.| } holds K1 is closed proof set K10 = [#] (TOP-REAL 2); reconsider g0 = (2 NormF) | ([#] (TOP-REAL 2)) as continuous Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by Lm5; reconsider g1 = proj2 | ([#] (TOP-REAL 2)) as continuous Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by Lm3; let sn be Real; ::_thesis: for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 >= sn * |.p7.| } holds K1 is closed let K1 be Subset of (TOP-REAL 2); ::_thesis: ( K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 >= sn * |.p7.| } implies K1 is closed ) defpred S1[ Point of (TOP-REAL 2)] means $1 `2 >= sn * |.$1.|; consider g2 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A1: for q being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1 being real number st g0 . q = r1 holds g2 . q = sn * r1 and A2: g2 is continuous by JGRAPH_2:23; consider g3 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A3: for q being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r1 - r2 and A4: g3 is continuous by A2, JGRAPH_2:21; A5: (TOP-REAL 2) | ([#] (TOP-REAL 2)) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) by TSEP_1:93; then reconsider g = g3 as Function of (TOP-REAL 2),R^1 ; reconsider K2 = K1 as Subset of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) ; assume K1 = { p where p is Point of (TOP-REAL 2) : p `2 >= sn * |.p.| } ; ::_thesis: K1 is closed then A6: K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } ; A7: K1 ` = { p7 where p7 is Point of (TOP-REAL 2) : not S1[p7] } from JGRAPH_2:sch_2(A6); A8: for p being Point of (TOP-REAL 2) holds g3 . p = (sn * |.p.|) - (p `2) proof let p be Point of (TOP-REAL 2); ::_thesis: g3 . p = (sn * |.p.|) - (p `2) g0 . p = (2 NormF) . p by A5, Lm5 .= |.p.| by Def1 ; then A9: g2 . p = sn * |.p.| by A1, A5; g1 . p = proj2 . p by A5, Lm3 .= p `2 by PSCOMP_1:def_6 ; hence g3 . p = (sn * |.p.|) - (p `2) by A3, A5, A9; ::_thesis: verum end; A10: K1 ` c= { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 ` or x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } ) assume x in K1 ` ; ::_thesis: x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } then consider p9 being Point of (TOP-REAL 2) such that A11: x = p9 and A12: p9 `2 < sn * |.p9.| by A7; A13: g /. p9 = (sn * |.p9.|) - (p9 `2) by A8; (sn * |.p9.|) - (p9 `2) > 0 by A12, XREAL_1:50; hence x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } by A11, A13; ::_thesis: verum end; { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } c= K1 ` proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } or x in K1 ` ) assume x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } ; ::_thesis: x in K1 ` then consider p7 being Point of (TOP-REAL 2) such that A14: p7 = x and A15: g /. p7 > 0 ; g /. p7 = (sn * |.p7.|) - (p7 `2) by A8; then ((sn * |.p7.|) - (p7 `2)) + (p7 `2) > 0 + (p7 `2) by A15, XREAL_1:8; hence x in K1 ` by A7, A14; ::_thesis: verum end; then K1 ` = { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } by A10, XBOOLE_0:def_10; then K2 ` is open by A4, A5, Th1; then K1 ` is open by PRE_TOPC:30; hence K1 is closed by TOPS_1:3; ::_thesis: verum end; Lm8: for sn being Real for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 >= sn * |.p7.| } holds K1 is closed proof set K10 = [#] (TOP-REAL 2); reconsider g0 = (2 NormF) | ([#] (TOP-REAL 2)) as continuous Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by Lm5; reconsider g1 = proj1 | ([#] (TOP-REAL 2)) as continuous Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by Lm2; let sn be Real; ::_thesis: for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 >= sn * |.p7.| } holds K1 is closed let K1 be Subset of (TOP-REAL 2); ::_thesis: ( K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 >= sn * |.p7.| } implies K1 is closed ) defpred S1[ Point of (TOP-REAL 2)] means $1 `1 >= sn * |.$1.|; consider g2 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A1: for q being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1 being real number st g0 . q = r1 holds g2 . q = sn * r1 and A2: g2 is continuous by JGRAPH_2:23; consider g3 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A3: for q being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r1 - r2 and A4: g3 is continuous by A2, JGRAPH_2:21; A5: (TOP-REAL 2) | ([#] (TOP-REAL 2)) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) by TSEP_1:93; then reconsider g = g3 as Function of (TOP-REAL 2),R^1 ; reconsider K2 = K1 as Subset of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) ; assume K1 = { p where p is Point of (TOP-REAL 2) : p `1 >= sn * |.p.| } ; ::_thesis: K1 is closed then A6: K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } ; A7: K1 ` = { p7 where p7 is Point of (TOP-REAL 2) : not S1[p7] } from JGRAPH_2:sch_2(A6); A8: for p being Point of (TOP-REAL 2) holds g3 . p = (sn * |.p.|) - (p `1) proof let p be Point of (TOP-REAL 2); ::_thesis: g3 . p = (sn * |.p.|) - (p `1) g0 . p = (2 NormF) . p by A5, Lm5 .= |.p.| by Def1 ; then A9: g2 . p = sn * |.p.| by A1, A5; g1 . p = proj1 . p by A5, Lm2 .= p `1 by PSCOMP_1:def_5 ; hence g3 . p = (sn * |.p.|) - (p `1) by A3, A5, A9; ::_thesis: verum end; A10: K1 ` c= { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 ` or x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } ) assume x in K1 ` ; ::_thesis: x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } then consider p9 being Point of (TOP-REAL 2) such that A11: x = p9 and A12: p9 `1 < sn * |.p9.| by A7; A13: g /. p9 = (sn * |.p9.|) - (p9 `1) by A8; (sn * |.p9.|) - (p9 `1) > 0 by A12, XREAL_1:50; hence x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } by A11, A13; ::_thesis: verum end; { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } c= K1 ` proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } or x in K1 ` ) assume x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } ; ::_thesis: x in K1 ` then consider p7 being Point of (TOP-REAL 2) such that A14: p7 = x and A15: g /. p7 > 0 ; g /. p7 = (sn * |.p7.|) - (p7 `1) by A8; then ((sn * |.p7.|) - (p7 `1)) + (p7 `1) > 0 + (p7 `1) by A15, XREAL_1:8; hence x in K1 ` by A7, A14; ::_thesis: verum end; then K1 ` = { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 > 0 } by A10, XBOOLE_0:def_10; then K2 ` is open by A4, A5, Th1; then K1 ` is open by PRE_TOPC:30; hence K1 is closed by TOPS_1:3; ::_thesis: verum end; theorem Th25: :: JGRAPH_4:25 for sn being Real for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 <= 0 ) } holds K03 is closed proof defpred S1[ Point of (TOP-REAL 2)] means $1 `1 <= 0 ; let sn be Real; ::_thesis: for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 <= 0 ) } holds K03 is closed let K003 be Subset of (TOP-REAL 2); ::_thesis: ( K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 <= 0 ) } implies K003 is closed ) defpred S2[ Point of (TOP-REAL 2)] means $1 `2 >= sn * |.$1.|; assume A1: K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 <= 0 ) } ; ::_thesis: K003 is closed reconsider KX = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A2: { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ { p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } from DOMAIN_1:sch_10(); ( K1 is closed & KX is closed ) by Lm7, JORDAN6:5; hence K003 is closed by A1, A2, TOPS_1:8; ::_thesis: verum end; Lm9: for sn being Real for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= sn * |.p7.| } holds K1 is closed proof set K10 = [#] (TOP-REAL 2); reconsider g0 = (2 NormF) | ([#] (TOP-REAL 2)) as continuous Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by Lm5; reconsider g1 = proj2 | ([#] (TOP-REAL 2)) as continuous Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by Lm3; let sn be Real; ::_thesis: for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= sn * |.p7.| } holds K1 is closed let K1 be Subset of (TOP-REAL 2); ::_thesis: ( K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= sn * |.p7.| } implies K1 is closed ) defpred S1[ Point of (TOP-REAL 2)] means $1 `2 <= sn * |.$1.|; consider g2 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A1: for q being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1 being real number st g0 . q = r1 holds g2 . q = sn * r1 and A2: g2 is continuous by JGRAPH_2:23; consider g3 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A3: for q being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r1 - r2 and A4: g3 is continuous by A2, JGRAPH_2:21; A5: (TOP-REAL 2) | ([#] (TOP-REAL 2)) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) by TSEP_1:93; then reconsider g = g3 as Function of (TOP-REAL 2),R^1 ; reconsider K2 = K1 as Subset of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) ; assume K1 = { p where p is Point of (TOP-REAL 2) : p `2 <= sn * |.p.| } ; ::_thesis: K1 is closed then A6: K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } ; A7: K1 ` = { p7 where p7 is Point of (TOP-REAL 2) : not S1[p7] } from JGRAPH_2:sch_2(A6); A8: for p being Point of (TOP-REAL 2) holds g3 . p = (sn * |.p.|) - (p `2) proof let p be Point of (TOP-REAL 2); ::_thesis: g3 . p = (sn * |.p.|) - (p `2) g0 . p = (2 NormF) . p by A5, Lm5 .= |.p.| by Def1 ; then A9: g2 . p = sn * |.p.| by A1, A5; g1 . p = proj2 . p by A5, Lm3 .= p `2 by PSCOMP_1:def_6 ; hence g3 . p = (sn * |.p.|) - (p `2) by A3, A5, A9; ::_thesis: verum end; A10: K1 ` c= { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 ` or x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } ) assume x in K1 ` ; ::_thesis: x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } then consider p9 being Point of (TOP-REAL 2) such that A11: x = p9 and A12: p9 `2 > sn * |.p9.| by A7; A13: g /. p9 = (sn * |.p9.|) - (p9 `2) by A8; (sn * |.p9.|) - (p9 `2) < 0 by A12, XREAL_1:49; hence x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } by A11, A13; ::_thesis: verum end; { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } c= K1 ` proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } or x in K1 ` ) assume x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } ; ::_thesis: x in K1 ` then consider p7 being Point of (TOP-REAL 2) such that A14: p7 = x and A15: g /. p7 < 0 ; g /. p7 = (sn * |.p7.|) - (p7 `2) by A8; then ((sn * |.p7.|) - (p7 `2)) + (p7 `2) < 0 + (p7 `2) by A15, XREAL_1:8; hence x in K1 ` by A7, A14; ::_thesis: verum end; then K1 ` = { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } by A10, XBOOLE_0:def_10; then K2 ` is open by A4, A5, Th2; then K1 ` is open by PRE_TOPC:30; hence K1 is closed by TOPS_1:3; ::_thesis: verum end; Lm10: for sn being Real for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= sn * |.p7.| } holds K1 is closed proof set K10 = [#] (TOP-REAL 2); reconsider g0 = (2 NormF) | ([#] (TOP-REAL 2)) as continuous Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by Lm5; reconsider g1 = proj1 | ([#] (TOP-REAL 2)) as continuous Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by Lm2; let sn be Real; ::_thesis: for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= sn * |.p7.| } holds K1 is closed let K1 be Subset of (TOP-REAL 2); ::_thesis: ( K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= sn * |.p7.| } implies K1 is closed ) defpred S1[ Point of (TOP-REAL 2)] means $1 `1 <= sn * |.$1.|; consider g2 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A1: for q being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1 being real number st g0 . q = r1 holds g2 . q = sn * r1 and A2: g2 is continuous by JGRAPH_2:23; consider g3 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that A3: for q being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2))) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r1 - r2 and A4: g3 is continuous by A2, JGRAPH_2:21; A5: (TOP-REAL 2) | ([#] (TOP-REAL 2)) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) by TSEP_1:93; then reconsider g = g3 as Function of (TOP-REAL 2),R^1 ; reconsider K2 = K1 as Subset of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) ; assume K1 = { p where p is Point of (TOP-REAL 2) : p `1 <= sn * |.p.| } ; ::_thesis: K1 is closed then A6: K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } ; A7: K1 ` = { p7 where p7 is Point of (TOP-REAL 2) : not S1[p7] } from JGRAPH_2:sch_2(A6); A8: for p being Point of (TOP-REAL 2) holds g3 . p = (sn * |.p.|) - (p `1) proof let p be Point of (TOP-REAL 2); ::_thesis: g3 . p = (sn * |.p.|) - (p `1) g0 . p = (2 NormF) . p by A5, Lm5 .= |.p.| by Def1 ; then A9: g2 . p = sn * |.p.| by A1, A5; g1 . p = proj1 . p by A5, Lm2 .= p `1 by PSCOMP_1:def_5 ; hence g3 . p = (sn * |.p.|) - (p `1) by A3, A5, A9; ::_thesis: verum end; A10: K1 ` c= { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 ` or x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } ) assume x in K1 ` ; ::_thesis: x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } then consider p9 being Point of (TOP-REAL 2) such that A11: x = p9 and A12: p9 `1 > sn * |.p9.| by A7; A13: g /. p9 = (sn * |.p9.|) - (p9 `1) by A8; (sn * |.p9.|) - (p9 `1) < 0 by A12, XREAL_1:49; hence x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } by A11, A13; ::_thesis: verum end; { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } c= K1 ` proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } or x in K1 ` ) assume x in { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } ; ::_thesis: x in K1 ` then consider p7 being Point of (TOP-REAL 2) such that A14: p7 = x and A15: g /. p7 < 0 ; g /. p7 = (sn * |.p7.|) - (p7 `1) by A8; then ((sn * |.p7.|) - (p7 `1)) + (p7 `1) < 0 + (p7 `1) by A15, XREAL_1:8; hence x in K1 ` by A7, A14; ::_thesis: verum end; then K1 ` = { p7 where p7 is Point of (TOP-REAL 2) : g /. p7 < 0 } by A10, XBOOLE_0:def_10; then K2 ` is open by A4, A5, Th2; then K1 ` is open by PRE_TOPC:30; hence K1 is closed by TOPS_1:3; ::_thesis: verum end; theorem Th26: :: JGRAPH_4:26 for sn being Real for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 <= 0 ) } holds K03 is closed proof defpred S1[ Point of (TOP-REAL 2)] means $1 `1 <= 0 ; let sn be Real; ::_thesis: for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 <= 0 ) } holds K03 is closed let K003 be Subset of (TOP-REAL 2); ::_thesis: ( K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 <= 0 ) } implies K003 is closed ) defpred S2[ Point of (TOP-REAL 2)] means $1 `2 <= sn * |.$1.|; assume A1: K003 = { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } ; ::_thesis: K003 is closed reconsider KX = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A2: { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ { p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } from DOMAIN_1:sch_10(); ( K1 is closed & KX is closed ) by Lm9, JORDAN6:5; hence K003 is closed by A1, A2, TOPS_1:8; ::_thesis: verum end; theorem Th27: :: JGRAPH_4:27 for sn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set cn = sqrt (1 - (sn ^2)); set p0 = |[(- (sqrt (1 - (sn ^2)))),sn]|; A1: |[(- (sqrt (1 - (sn ^2)))),sn]| `1 = - (sqrt (1 - (sn ^2))) by EUCLID:52; |[(- (sqrt (1 - (sn ^2)))),sn]| `2 = sn by EUCLID:52; then A2: |.|[(- (sqrt (1 - (sn ^2)))),sn]|.| = sqrt (((- (sqrt (1 - (sn ^2)))) ^2) + (sn ^2)) by A1, JGRAPH_3:1 .= sqrt (((sqrt (1 - (sn ^2))) ^2) + (sn ^2)) ; assume A3: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then sn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (sn ^2) > 0 by XREAL_1:50; then A5: - (- (sqrt (1 - (sn ^2)))) > 0 by SQUARE_1:25; A6: now__::_thesis:_not_|[(-_(sqrt_(1_-_(sn_^2)))),sn]|_=_0._(TOP-REAL_2) assume |[(- (sqrt (1 - (sn ^2)))),sn]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then - (- (sqrt (1 - (sn ^2)))) = - 0 by EUCLID:52, JGRAPH_2:3; hence contradiction by A4, SQUARE_1:25; ::_thesis: verum end; then |[(- (sqrt (1 - (sn ^2)))),sn]| in K0 by A3, A1, A5; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; (sqrt (1 - (sn ^2))) ^2 = 1 - (sn ^2) by A4, SQUARE_1:def_2; then A7: (|[(- (sqrt (1 - (sn ^2)))),sn]| `2) / |.|[(- (sqrt (1 - (sn ^2)))),sn]|.| = sn by A2, EUCLID:52, SQUARE_1:18; then A8: |[(- (sqrt (1 - (sn ^2)))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5; not |[(- (sqrt (1 - (sn ^2)))),sn]| in {(0. (TOP-REAL 2))} by A6, TARSKI:def_1; then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A3, XBOOLE_0:def_5; K1 c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in D ) assume A9: x in K1 ; ::_thesis: x in D then ex p6 being Point of (TOP-REAL 2) st ( p6 = x & p6 `1 <= 0 & p6 <> 0. (TOP-REAL 2) ) by A3; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in D by A3, A9, XBOOLE_0:def_5; ::_thesis: verum end; then D = K1 \/ D by XBOOLE_1:12; then A10: (TOP-REAL 2) | K1 is SubSpace of (TOP-REAL 2) | D by TOPMETR:4; A11: { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } c= K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 ) assume x in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in K1 then ex p being Point of (TOP-REAL 2) st ( p = x & (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) ; hence x in K1 by A3; ::_thesis: verum end; A12: { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } c= K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 ) assume x in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in K1 then ex p being Point of (TOP-REAL 2) st ( p = x & (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) ; hence x in K1 by A3; ::_thesis: verum end; then reconsider K00 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A8, PRE_TOPC:8; the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; then A13: rng (f | K00) c= D ; |[(- (sqrt (1 - (sn ^2)))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5, A7; then reconsider K11 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A11, PRE_TOPC:8; the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; then A14: rng (f | K11) c= D ; the carrier of ((TOP-REAL 2) | B0) = the carrier of ((TOP-REAL 2) | D) ; then A15: dom f = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1 .= K1 by PRE_TOPC:8 ; then dom (f | K00) = K00 by A12, RELAT_1:62 .= the carrier of (((TOP-REAL 2) | K1) | K00) by PRE_TOPC:8 ; then reconsider f1 = f | K00 as Function of (((TOP-REAL 2) | K1) | K00),((TOP-REAL 2) | D) by A13, FUNCT_2:2; dom (f | K11) = K11 by A11, A15, RELAT_1:62 .= the carrier of (((TOP-REAL 2) | K1) | K11) by PRE_TOPC:8 ; then reconsider f2 = f | K11 as Function of (((TOP-REAL 2) | K1) | K11),((TOP-REAL 2) | D) by A14, FUNCT_2:2; A16: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; defpred S1[ Point of (TOP-REAL 2)] means ( ($1 `2) / |.$1.| >= sn & $1 `1 <= 0 & $1 <> 0. (TOP-REAL 2) ); A17: dom f2 = the carrier of (((TOP-REAL 2) | K1) | K11) by FUNCT_2:def_1 .= K11 by PRE_TOPC:8 ; { p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K001 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A8; A18: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; defpred S2[ Point of (TOP-REAL 2)] means ( $1 `2 >= sn * |.$1.| & $1 `1 <= 0 ); { p where p is Point of (TOP-REAL 2) : S2[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 <= 0 ) } as Subset of (TOP-REAL 2) ; defpred S3[ Point of (TOP-REAL 2)] means ( ($1 `2) / |.$1.| <= sn & $1 `1 <= 0 & $1 <> 0. (TOP-REAL 2) ); A19: { p where p is Point of (TOP-REAL 2) : S3[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); A20: rng ((sn -FanMorphW) | K001) c= K1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((sn -FanMorphW) | K001) or y in K1 ) assume y in rng ((sn -FanMorphW) | K001) ; ::_thesis: y in K1 then consider x being set such that A21: x in dom ((sn -FanMorphW) | K001) and A22: y = ((sn -FanMorphW) | K001) . x by FUNCT_1:def_3; x in dom (sn -FanMorphW) by A21, RELAT_1:57; then reconsider q = x as Point of (TOP-REAL 2) ; A23: y = (sn -FanMorphW) . q by A21, A22, FUNCT_1:47; dom ((sn -FanMorphW) | K001) = (dom (sn -FanMorphW)) /\ K001 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K001 by FUNCT_2:def_1 .= K001 by XBOOLE_1:28 ; then A24: ex p2 being Point of (TOP-REAL 2) st ( p2 = q & (p2 `2) / |.p2.| >= sn & p2 `1 <= 0 & p2 <> 0. (TOP-REAL 2) ) by A21; then A25: ((q `2) / |.q.|) - sn >= 0 by XREAL_1:48; |.q.| <> 0 by A24, TOPRNS_1:24; then A26: |.q.| ^2 > 0 ^2 by SQUARE_1:12; set q4 = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]|; A27: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)) by EUCLID:52; A28: 1 - sn > 0 by A3, XREAL_1:149; 0 <= (q `1) ^2 by XREAL_1:63; then 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then (q `2) ^2 <= |.q.| ^2 by JGRAPH_3:1; then ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A26, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `2) / |.q.| by SQUARE_1:51; then 1 - sn >= ((q `2) / |.q.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((q `2) / |.q.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A28, XREAL_1:72; then - 1 <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A28, XCMPLX_1:197; then ((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A28, A25, SQUARE_1:49; then A29: 1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A30: 1 - ((- ((((q `2) / |.q.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; sqrt (1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2)) >= 0 by A29, SQUARE_1:def_2; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) ^2) / ((1 - sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((q `2) / |.q.|) - sn) ^2) / ((1 - sn) ^2))) >= 0 ; then A31: sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) >= 0 by XCMPLX_1:76; A32: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; then A33: (|[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) by A30, SQUARE_1:def_2 ; |.|[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]|.| ^2 = ((|[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `1) ^2) + ((|[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A27, A33 ; then A34: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| <> 0. (TOP-REAL 2) by A26, TOPRNS_1:23; (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by A3, A24, Th18; hence y in K1 by A3, A23, A32, A31, A34; ::_thesis: verum end; A35: dom (sn -FanMorphW) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then dom ((sn -FanMorphW) | K001) = K001 by RELAT_1:62 .= the carrier of ((TOP-REAL 2) | K001) by PRE_TOPC:8 ; then reconsider f3 = (sn -FanMorphW) | K001 as Function of ((TOP-REAL 2) | K001),((TOP-REAL 2) | K1) by A18, A20, FUNCT_2:2; A36: K003 is closed by Th25; defpred S4[ Point of (TOP-REAL 2)] means ( $1 `2 <= sn * |.$1.| & $1 `1 <= 0 ); { p where p is Point of (TOP-REAL 2) : S4[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K004 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 <= 0 ) } as Subset of (TOP-REAL 2) ; A37: K004 /\ K1 c= K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K004 /\ K1 or x in K11 ) assume A38: x in K004 /\ K1 ; ::_thesis: x in K11 then x in K004 by XBOOLE_0:def_4; then consider q1 being Point of (TOP-REAL 2) such that A39: q1 = x and A40: q1 `2 <= sn * |.q1.| and q1 `1 <= 0 ; x in K1 by A38, XBOOLE_0:def_4; then A41: ex q2 being Point of (TOP-REAL 2) st ( q2 = x & q2 `1 <= 0 & q2 <> 0. (TOP-REAL 2) ) by A3; (q1 `2) / |.q1.| <= (sn * |.q1.|) / |.q1.| by A40, XREAL_1:72; then (q1 `2) / |.q1.| <= sn by A39, A41, TOPRNS_1:24, XCMPLX_1:89; hence x in K11 by A39, A41; ::_thesis: verum end; A42: K004 is closed by Th26; the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then ( ((TOP-REAL 2) | K1) | K00 = (TOP-REAL 2) | K001 & f1 = f3 ) by A3, FUNCT_1:51, GOBOARD9:2; then A43: f1 is continuous by A3, A10, Th23, PRE_TOPC:26; A44: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def_5; |[(- (sqrt (1 - (sn ^2)))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5, A7; then reconsider K111 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A19; A45: rng ((sn -FanMorphW) | K111) c= K1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((sn -FanMorphW) | K111) or y in K1 ) assume y in rng ((sn -FanMorphW) | K111) ; ::_thesis: y in K1 then consider x being set such that A46: x in dom ((sn -FanMorphW) | K111) and A47: y = ((sn -FanMorphW) | K111) . x by FUNCT_1:def_3; x in dom (sn -FanMorphW) by A46, RELAT_1:57; then reconsider q = x as Point of (TOP-REAL 2) ; A48: y = (sn -FanMorphW) . q by A46, A47, FUNCT_1:47; dom ((sn -FanMorphW) | K111) = (dom (sn -FanMorphW)) /\ K111 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K111 by FUNCT_2:def_1 .= K111 by XBOOLE_1:28 ; then A49: ex p2 being Point of (TOP-REAL 2) st ( p2 = q & (p2 `2) / |.p2.| <= sn & p2 `1 <= 0 & p2 <> 0. (TOP-REAL 2) ) by A46; then A50: ((q `2) / |.q.|) - sn <= 0 by XREAL_1:47; |.q.| <> 0 by A49, TOPRNS_1:24; then A51: |.q.| ^2 > 0 ^2 by SQUARE_1:12; set q4 = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|; A52: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) by EUCLID:52; A53: 1 + sn > 0 by A3, XREAL_1:148; 0 <= (q `1) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A51, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `2) / |.q.| by SQUARE_1:51; then (- 1) - sn <= ((q `2) / |.q.|) - sn by XREAL_1:9; then (- (1 + sn)) / (1 + sn) <= (((q `2) / |.q.|) - sn) / (1 + sn) by A53, XREAL_1:72; then - 1 <= (((q `2) / |.q.|) - sn) / (1 + sn) by A53, XCMPLX_1:197; then A54: ((((q `2) / |.q.|) - sn) / (1 + sn)) ^2 <= 1 ^2 by A53, A50, SQUARE_1:49; then A55: 1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2) >= 0 by XREAL_1:48; 1 - ((- ((((q `2) / |.q.|) - sn) / (1 + sn))) ^2) >= 0 by A54, XREAL_1:48; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2) >= 0 by XCMPLX_1:187; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) ^2) / ((1 + sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((q `2) / |.q.|) - sn) ^2) / ((1 + sn) ^2))) >= 0 ; then A56: sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) >= 0 by XCMPLX_1:76; A57: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))) by EUCLID:52; then A58: (|[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) by A55, SQUARE_1:def_2 ; |.|[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|.| ^2 = ((|[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `1) ^2) + ((|[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A52, A58 ; then A59: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| <> 0. (TOP-REAL 2) by A51, TOPRNS_1:23; (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A3, A49, Th18; hence y in K1 by A3, A48, A57, A56, A59; ::_thesis: verum end; dom ((sn -FanMorphW) | K111) = K111 by A35, RELAT_1:62 .= the carrier of ((TOP-REAL 2) | K111) by PRE_TOPC:8 ; then reconsider f4 = (sn -FanMorphW) | K111 as Function of ((TOP-REAL 2) | K111),((TOP-REAL 2) | K1) by A16, A45, FUNCT_2:2; the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then ( ((TOP-REAL 2) | K1) | K11 = (TOP-REAL 2) | K111 & f2 = f4 ) by A3, FUNCT_1:51, GOBOARD9:2; then A60: f2 is continuous by A3, A10, Th24, PRE_TOPC:26; set T1 = ((TOP-REAL 2) | K1) | K00; set T2 = ((TOP-REAL 2) | K1) | K11; A61: [#] (((TOP-REAL 2) | K1) | K11) = K11 by PRE_TOPC:def_5; K11 c= K004 /\ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K11 or x in K004 /\ K1 ) assume x in K11 ; ::_thesis: x in K004 /\ K1 then consider p being Point of (TOP-REAL 2) such that A62: p = x and A63: (p `2) / |.p.| <= sn and A64: p `1 <= 0 and A65: p <> 0. (TOP-REAL 2) ; ((p `2) / |.p.|) * |.p.| <= sn * |.p.| by A63, XREAL_1:64; then p `2 <= sn * |.p.| by A65, TOPRNS_1:24, XCMPLX_1:87; then A66: x in K004 by A62, A64; x in K1 by A3, A62, A64, A65; hence x in K004 /\ K1 by A66, XBOOLE_0:def_4; ::_thesis: verum end; then K11 = K004 /\ ([#] ((TOP-REAL 2) | K1)) by A44, A37, XBOOLE_0:def_10; then A67: K11 is closed by A42, PRE_TOPC:13; A68: K003 /\ K1 c= K00 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K003 /\ K1 or x in K00 ) assume A69: x in K003 /\ K1 ; ::_thesis: x in K00 then x in K003 by XBOOLE_0:def_4; then consider q1 being Point of (TOP-REAL 2) such that A70: q1 = x and A71: q1 `2 >= sn * |.q1.| and q1 `1 <= 0 ; x in K1 by A69, XBOOLE_0:def_4; then A72: ex q2 being Point of (TOP-REAL 2) st ( q2 = x & q2 `1 <= 0 & q2 <> 0. (TOP-REAL 2) ) by A3; (q1 `2) / |.q1.| >= (sn * |.q1.|) / |.q1.| by A71, XREAL_1:72; then (q1 `2) / |.q1.| >= sn by A70, A72, TOPRNS_1:24, XCMPLX_1:89; hence x in K00 by A70, A72; ::_thesis: verum end; A73: the carrier of ((TOP-REAL 2) | K1) = K0 by PRE_TOPC:8; A74: D <> {} ; A75: [#] (((TOP-REAL 2) | K1) | K00) = K00 by PRE_TOPC:def_5; A76: for p being set st p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) holds f1 . p = f2 . p proof let p be set ; ::_thesis: ( p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) implies f1 . p = f2 . p ) assume A77: p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) ; ::_thesis: f1 . p = f2 . p then p in K00 by A75, XBOOLE_0:def_4; hence f1 . p = f . p by FUNCT_1:49 .= f2 . p by A61, A77, FUNCT_1:49 ; ::_thesis: verum end; K00 c= K003 /\ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K00 or x in K003 /\ K1 ) assume x in K00 ; ::_thesis: x in K003 /\ K1 then consider p being Point of (TOP-REAL 2) such that A78: p = x and A79: (p `2) / |.p.| >= sn and A80: p `1 <= 0 and A81: p <> 0. (TOP-REAL 2) ; ((p `2) / |.p.|) * |.p.| >= sn * |.p.| by A79, XREAL_1:64; then p `2 >= sn * |.p.| by A81, TOPRNS_1:24, XCMPLX_1:87; then A82: x in K003 by A78, A80; x in K1 by A3, A78, A80, A81; hence x in K003 /\ K1 by A82, XBOOLE_0:def_4; ::_thesis: verum end; then K00 = K003 /\ ([#] ((TOP-REAL 2) | K1)) by A44, A68, XBOOLE_0:def_10; then A83: K00 is closed by A36, PRE_TOPC:13; A84: K1 c= K00 \/ K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in K00 \/ K11 ) assume x in K1 ; ::_thesis: x in K00 \/ K11 then consider p being Point of (TOP-REAL 2) such that A85: ( p = x & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) by A3; percases ( (p `2) / |.p.| >= sn or (p `2) / |.p.| < sn ) ; suppose (p `2) / |.p.| >= sn ; ::_thesis: x in K00 \/ K11 then x in K00 by A85; hence x in K00 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; suppose (p `2) / |.p.| < sn ; ::_thesis: x in K00 \/ K11 then x in K11 by A85; hence x in K00 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; end; end; then ([#] (((TOP-REAL 2) | K1) | K00)) \/ ([#] (((TOP-REAL 2) | K1) | K11)) = [#] ((TOP-REAL 2) | K1) by A75, A61, A44, XBOOLE_0:def_10; then consider h being Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) such that A86: h = f1 +* f2 and A87: h is continuous by A75, A61, A83, A67, A43, A60, A76, JGRAPH_2:1; A88: dom h = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; A89: dom f1 = the carrier of (((TOP-REAL 2) | K1) | K00) by FUNCT_2:def_1 .= K00 by PRE_TOPC:8 ; A90: for y being set st y in dom h holds h . y = f . y proof let y be set ; ::_thesis: ( y in dom h implies h . y = f . y ) assume A91: y in dom h ; ::_thesis: h . y = f . y now__::_thesis:_(_(_y_in_K00_&_not_y_in_K11_&_h_._y_=_f_._y_)_or_(_y_in_K11_&_h_._y_=_f_._y_)_) percases ( ( y in K00 & not y in K11 ) or y in K11 ) by A84, A88, A73, A91, XBOOLE_0:def_3; caseA92: ( y in K00 & not y in K11 ) ; ::_thesis: h . y = f . y then y in (dom f1) \/ (dom f2) by A89, XBOOLE_0:def_3; hence h . y = f1 . y by A17, A86, A92, FUNCT_4:def_1 .= f . y by A92, FUNCT_1:49 ; ::_thesis: verum end; caseA93: y in K11 ; ::_thesis: h . y = f . y then y in (dom f1) \/ (dom f2) by A17, XBOOLE_0:def_3; hence h . y = f2 . y by A17, A86, A93, FUNCT_4:def_1 .= f . y by A93, FUNCT_1:49 ; ::_thesis: verum end; end; end; hence h . y = f . y ; ::_thesis: verum end; K0 = the carrier of ((TOP-REAL 2) | K0) by PRE_TOPC:8 .= dom f by A74, FUNCT_2:def_1 ; hence f is continuous by A87, A88, A90, FUNCT_1:2, PRE_TOPC:8; ::_thesis: verum end; theorem Th28: :: JGRAPH_4:28 for sn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set cn = sqrt (1 - (sn ^2)); set p0 = |[(sqrt (1 - (sn ^2))),(- sn)]|; assume A1: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then sn ^2 < 1 ^2 by SQUARE_1:50; then A2: ( |[(sqrt (1 - (sn ^2))),(- sn)]| `1 = sqrt (1 - (sn ^2)) & 1 - (sn ^2) > 0 ) by EUCLID:52, XREAL_1:50; then |[(sqrt (1 - (sn ^2))),(- sn)]| <> 0. (TOP-REAL 2) by JGRAPH_2:3, SQUARE_1:25; then not |[(sqrt (1 - (sn ^2))),(- sn)]| in {(0. (TOP-REAL 2))} by TARSKI:def_1; then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A1, XBOOLE_0:def_5; |[(sqrt (1 - (sn ^2))),(- sn)]| `1 > 0 by A2, SQUARE_1:25; then |[(sqrt (1 - (sn ^2))),(- sn)]| in K0 by A1, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A3: K1 c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in D ) assume x in K1 ; ::_thesis: x in D then consider p2 being Point of (TOP-REAL 2) such that A4: p2 = x and p2 `1 >= 0 and A5: p2 <> 0. (TOP-REAL 2) by A1; not p2 in {(0. (TOP-REAL 2))} by A5, TARSKI:def_1; hence x in D by A1, A4, XBOOLE_0:def_5; ::_thesis: verum end; for p being Point of ((TOP-REAL 2) | K1) for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) proof let p be Point of ((TOP-REAL 2) | K1); ::_thesis: for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) let V be Subset of ((TOP-REAL 2) | D); ::_thesis: ( f . p in V & V is open implies ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) ) assume that A6: f . p in V and A7: V is open ; ::_thesis: ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) consider V2 being Subset of (TOP-REAL 2) such that A8: V2 is open and A9: V2 /\ ([#] ((TOP-REAL 2) | D)) = V by A7, TOPS_2:24; reconsider W2 = V2 /\ ([#] ((TOP-REAL 2) | K1)) as Subset of ((TOP-REAL 2) | K1) ; A10: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def_5; then A11: f . p = (sn -FanMorphW) . p by A1, FUNCT_1:49; A12: f .: W2 c= V proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f .: W2 or y in V ) assume y in f .: W2 ; ::_thesis: y in V then consider x being set such that A13: x in dom f and A14: x in W2 and A15: y = f . x by FUNCT_1:def_6; f is Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) ; then dom f = K1 by A10, FUNCT_2:def_1; then consider p4 being Point of (TOP-REAL 2) such that A16: x = p4 and A17: p4 `1 >= 0 and p4 <> 0. (TOP-REAL 2) by A1, A13; A18: p4 in V2 by A14, A16, XBOOLE_0:def_4; p4 in [#] ((TOP-REAL 2) | K1) by A13, A16; then p4 in D by A3, A10; then A19: p4 in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; f . p4 = (sn -FanMorphW) . p4 by A1, A10, A13, A16, FUNCT_1:49 .= p4 by A17, Th16 ; hence y in V by A9, A15, A16, A18, A19, XBOOLE_0:def_4; ::_thesis: verum end; p in the carrier of ((TOP-REAL 2) | K1) ; then consider q being Point of (TOP-REAL 2) such that A20: q = p and A21: q `1 >= 0 and q <> 0. (TOP-REAL 2) by A1, A10; (sn -FanMorphW) . q = q by A21, Th16; then p in V2 by A6, A9, A11, A20, XBOOLE_0:def_4; then A22: p in W2 by XBOOLE_0:def_4; W2 is open by A8, TOPS_2:24; hence ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) by A22, A12; ::_thesis: verum end; hence f is continuous by JGRAPH_2:10; ::_thesis: verum end; theorem Th29: :: JGRAPH_4:29 for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds K0 is closed proof set J0 = NonZero (TOP-REAL 2); defpred S1[ Point of (TOP-REAL 2)] means $1 `1 <= 0 ; set I1 = { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } ; let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds K0 is closed let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies K0 is closed ) reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A1: { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch_2(); assume ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: K0 is closed then ( K1 is closed & K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) ) by A1, JORDAN6:5, PRE_TOPC:def_5; hence K0 is closed by PRE_TOPC:13; ::_thesis: verum end; theorem Th30: :: JGRAPH_4:30 for sn being Real for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof defpred S1[ Point of (TOP-REAL 2)] means ( $1 `1 <= 0 & $1 <> 0. (TOP-REAL 2) ); let sn be Real; ::_thesis: for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) reconsider K1 = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); assume A1: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & p8 `1 <= 0 & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by A1, PRE_TOPC:7; hence f is continuous by A1, Th27; ::_thesis: verum end; theorem Th31: :: JGRAPH_4:31 for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds K0 is closed proof set J0 = NonZero (TOP-REAL 2); defpred S1[ Point of (TOP-REAL 2)] means $1 `1 >= 0 ; set I1 = { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } ; let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds K0 is closed let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies K0 is closed ) reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A1: { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch_2(); assume ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: K0 is closed then ( K1 is closed & K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) ) by A1, JORDAN6:4, PRE_TOPC:def_5; hence K0 is closed by PRE_TOPC:13; ::_thesis: verum end; theorem Th32: :: JGRAPH_4:32 for sn being Real for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:8; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & p8 `1 >= 0 & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; hence f is continuous by A1, Th28; ::_thesis: verum end; theorem Th33: :: JGRAPH_4:33 for sn being Real for p being Point of (TOP-REAL 2) holds |.((sn -FanMorphW) . p).| = |.p.| proof let sn be Real; ::_thesis: for p being Point of (TOP-REAL 2) holds |.((sn -FanMorphW) . p).| = |.p.| let p be Point of (TOP-REAL 2); ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| set z = (sn -FanMorphW) . p; reconsider q = p, qz = (sn -FanMorphW) . p as Point of (TOP-REAL 2) ; percases ( ( (q `2) / |.q.| >= sn & q `1 < 0 ) or ( (q `2) / |.q.| < sn & q `1 < 0 ) or q `1 >= 0 ) ; supposeA1: ( (q `2) / |.q.| >= sn & q `1 < 0 ) ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| then A2: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by Th16; then A3: qz `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; A4: qz `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)) by A2, EUCLID:52; A5: ((q `2) / |.q.|) - sn >= 0 by A1, XREAL_1:48; A6: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; |.q.| <> 0 by A1, JGRAPH_2:3, TOPRNS_1:24; then A7: |.q.| ^2 > 0 by SQUARE_1:12; 0 <= (q `1) ^2 by XREAL_1:63; then 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by A6, XREAL_1:72; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A7, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `2) / |.q.| by SQUARE_1:51; then A8: 1 - sn >= ((q `2) / |.q.|) - sn by XREAL_1:9; percases ( 1 - sn = 0 or 1 - sn <> 0 ) ; supposeA9: 1 - sn = 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| A10: (((q `2) / |.q.|) - sn) / (1 - sn) = (((q `2) / |.q.|) - sn) * ((1 - sn) ") by XCMPLX_0:def_9 .= (((q `2) / |.q.|) - sn) * 0 by A9 .= 0 ; then - (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) = - 1 by SQUARE_1:18; then (sn -FanMorphW) . q = |[(|.q.| * (- 1)),(|.q.| * 0)]| by A1, A10, Th16 .= |[(- |.q.|),0]| ; then ( ((sn -FanMorphW) . q) `1 = - |.q.| & ((sn -FanMorphW) . q) `2 = 0 ) by EUCLID:52; then |.((sn -FanMorphW) . p).| = sqrt (((- |.q.|) ^2) + (0 ^2)) by JGRAPH_3:1 .= sqrt (|.q.| ^2) .= |.q.| by SQUARE_1:22 ; hence |.((sn -FanMorphW) . p).| = |.p.| ; ::_thesis: verum end; supposeA11: 1 - sn <> 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| percases ( 1 - sn > 0 or 1 - sn < 0 ) by A11; supposeA12: 1 - sn > 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| - (1 - sn) <= - (((q `2) / |.q.|) - sn) by A8, XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A12, XREAL_1:72; then - 1 <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A12, XCMPLX_1:197; then ((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A5, A12, SQUARE_1:49; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A13: 1 - ((- ((((q `2) / |.q.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; A14: (qz `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) ^2) by A3 .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) by A13, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A4, A14 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; hence |.((sn -FanMorphW) . p).| = |.p.| by SQUARE_1:22; ::_thesis: verum end; supposeA15: 1 - sn < 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| 0 + ((q `2) ^2) < ((q `1) ^2) + ((q `2) ^2) by A1, SQUARE_1:12, XREAL_1:8; then ((q `2) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A7, A6, XREAL_1:74; then ((q `2) ^2) / (|.q.| ^2) < 1 by A7, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 < 1 by XCMPLX_1:76; then A16: 1 > (q `2) / |.p.| by SQUARE_1:52; ((q `2) / |.q.|) - sn >= 0 by A1, XREAL_1:48; hence |.((sn -FanMorphW) . p).| = |.p.| by A15, A16, XREAL_1:9; ::_thesis: verum end; end; end; end; end; supposeA17: ( (q `2) / |.q.| < sn & q `1 < 0 ) ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| then |.q.| <> 0 by JGRAPH_2:3, TOPRNS_1:24; then A18: |.q.| ^2 > 0 by SQUARE_1:12; A19: ((q `2) / |.q.|) - sn < 0 by A17, XREAL_1:49; A20: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; 0 <= (q `1) ^2 by XREAL_1:63; then 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by A20, XREAL_1:72; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A18, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `2) / |.q.| by SQUARE_1:51; then A21: (- 1) - sn <= ((q `2) / |.q.|) - sn by XREAL_1:9; A22: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A17, Th17; then A23: qz `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))) by EUCLID:52; A24: qz `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) by A22, EUCLID:52; percases ( 1 + sn = 0 or 1 + sn <> 0 ) ; supposeA25: 1 + sn = 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| (((q `2) / |.q.|) - sn) / (1 + sn) = (((q `2) / |.q.|) - sn) * ((1 + sn) ") by XCMPLX_0:def_9 .= (((q `2) / |.q.|) - sn) * 0 by A25 .= 0 ; then ( ((sn -FanMorphW) . q) `1 = - |.q.| & ((sn -FanMorphW) . q) `2 = 0 ) by A22, EUCLID:52, SQUARE_1:18; then |.((sn -FanMorphW) . p).| = sqrt (((- |.q.|) ^2) + (0 ^2)) by JGRAPH_3:1 .= sqrt (|.q.| ^2) .= |.q.| by SQUARE_1:22 ; hence |.((sn -FanMorphW) . p).| = |.p.| ; ::_thesis: verum end; supposeA26: 1 + sn <> 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| percases ( 1 + sn > 0 or 1 + sn < 0 ) by A26; supposeA27: 1 + sn > 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| then (- (1 + sn)) / (1 + sn) <= (((q `2) / |.q.|) - sn) / (1 + sn) by A21, XREAL_1:72; then - 1 <= (((q `2) / |.q.|) - sn) / (1 + sn) by A27, XCMPLX_1:197; then ((((q `2) / |.q.|) - sn) / (1 + sn)) ^2 <= 1 ^2 by A19, A27, SQUARE_1:49; then A28: 1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2) >= 0 by XREAL_1:48; A29: (qz `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) ^2) by A23 .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) by A28, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A24, A29 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; hence |.((sn -FanMorphW) . p).| = |.p.| by SQUARE_1:22; ::_thesis: verum end; supposeA30: 1 + sn < 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| 0 + ((q `2) ^2) < ((q `1) ^2) + ((q `2) ^2) by A17, SQUARE_1:12, XREAL_1:8; then ((q `2) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A18, A20, XREAL_1:74; then ((q `2) ^2) / (|.q.| ^2) < 1 by A18, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (q `2) / |.p.| by SQUARE_1:52; then A31: ((q `2) / |.q.|) - sn > (- 1) - sn by XREAL_1:9; - (1 + sn) > - 0 by A30, XREAL_1:24; hence |.((sn -FanMorphW) . p).| = |.p.| by A17, A31, XREAL_1:49; ::_thesis: verum end; end; end; end; end; suppose q `1 >= 0 ; ::_thesis: |.((sn -FanMorphW) . p).| = |.p.| hence |.((sn -FanMorphW) . p).| = |.p.| by Th16; ::_thesis: verum end; end; end; theorem Th34: :: JGRAPH_4:34 for sn being Real for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds (sn -FanMorphW) . x in K0 proof let sn be Real; ::_thesis: for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds (sn -FanMorphW) . x in K0 let x, K0 be set ; ::_thesis: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies (sn -FanMorphW) . x in K0 ) assume A1: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: (sn -FanMorphW) . x in K0 then consider p being Point of (TOP-REAL 2) such that A2: p = x and A3: p `1 <= 0 and A4: p <> 0. (TOP-REAL 2) ; A5: now__::_thesis:_not_|.p.|_<=_0 assume |.p.| <= 0 ; ::_thesis: contradiction then |.p.| = 0 ; hence contradiction by A4, TOPRNS_1:24; ::_thesis: verum end; then A6: |.p.| ^2 > 0 by SQUARE_1:12; percases ( (p `2) / |.p.| <= sn or (p `2) / |.p.| > sn ) ; supposeA7: (p `2) / |.p.| <= sn ; ::_thesis: (sn -FanMorphW) . x in K0 reconsider p9 = (sn -FanMorphW) . p as Point of (TOP-REAL 2) ; (sn -FanMorphW) . p = |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A1, A3, A4, A7, Th18; then A8: p9 `1 = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) by EUCLID:52; A9: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; A10: 1 + sn > 0 by A1, XREAL_1:148; percases ( p `1 = 0 or p `1 <> 0 ) ; suppose p `1 = 0 ; ::_thesis: (sn -FanMorphW) . x in K0 hence (sn -FanMorphW) . x in K0 by A1, A2, Th16; ::_thesis: verum end; suppose p `1 <> 0 ; ::_thesis: (sn -FanMorphW) . x in K0 then 0 + ((p `2) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8; then ((p `2) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A9, XREAL_1:74; then ((p `2) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60; then ((p `2) / |.p.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (p `2) / |.p.| by SQUARE_1:52; then (- 1) - sn < ((p `2) / |.p.|) - sn by XREAL_1:9; then ((- 1) * (1 + sn)) / (1 + sn) < (((p `2) / |.p.|) - sn) / (1 + sn) by A10, XREAL_1:74; then A11: - 1 < (((p `2) / |.p.|) - sn) / (1 + sn) by A10, XCMPLX_1:89; ((p `2) / |.p.|) - sn <= 0 by A7, XREAL_1:47; then 1 ^2 > ((((p `2) / |.p.|) - sn) / (1 + sn)) ^2 by A10, A11, SQUARE_1:50; then 1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2) > 0 by XREAL_1:50; then - (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) > 0 by SQUARE_1:25; then - (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) < 0 ; then |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) < 0 by A5, XREAL_1:132; hence (sn -FanMorphW) . x in K0 by A1, A2, A8, JGRAPH_2:3; ::_thesis: verum end; end; end; supposeA12: (p `2) / |.p.| > sn ; ::_thesis: (sn -FanMorphW) . x in K0 reconsider p9 = (sn -FanMorphW) . p as Point of (TOP-REAL 2) ; (sn -FanMorphW) . p = |[(|.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A1, A3, A4, A12, Th18; then A13: p9 `1 = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; A14: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; A15: 1 - sn > 0 by A1, XREAL_1:149; percases ( p `1 = 0 or p `1 <> 0 ) ; suppose p `1 = 0 ; ::_thesis: (sn -FanMorphW) . x in K0 hence (sn -FanMorphW) . x in K0 by A1, A2, Th16; ::_thesis: verum end; suppose p `1 <> 0 ; ::_thesis: (sn -FanMorphW) . x in K0 then 0 + ((p `2) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8; then ((p `2) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A14, XREAL_1:74; then ((p `2) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60; then ((p `2) / |.p.|) ^2 < 1 by XCMPLX_1:76; then (p `2) / |.p.| < 1 by SQUARE_1:52; then ((p `2) / |.p.|) - sn < 1 - sn by XREAL_1:9; then (((p `2) / |.p.|) - sn) / (1 - sn) < (1 - sn) / (1 - sn) by A15, XREAL_1:74; then A16: (((p `2) / |.p.|) - sn) / (1 - sn) < 1 by A15, XCMPLX_1:60; ( - (1 - sn) < - 0 & ((p `2) / |.p.|) - sn >= sn - sn ) by A12, A15, XREAL_1:9, XREAL_1:24; then ((- 1) * (1 - sn)) / (1 - sn) < (((p `2) / |.p.|) - sn) / (1 - sn) by A15, XREAL_1:74; then - 1 < (((p `2) / |.p.|) - sn) / (1 - sn) by A15, XCMPLX_1:89; then 1 ^2 > ((((p `2) / |.p.|) - sn) / (1 - sn)) ^2 by A16, SQUARE_1:50; then 1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2) > 0 by XREAL_1:50; then - (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) > 0 by SQUARE_1:25; then - (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) < 0 ; then p9 `1 < 0 by A5, A13, XREAL_1:132; hence (sn -FanMorphW) . x in K0 by A1, A2, JGRAPH_2:3; ::_thesis: verum end; end; end; end; end; theorem Th35: :: JGRAPH_4:35 for sn being Real for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds (sn -FanMorphW) . x in K0 proof let sn be Real; ::_thesis: for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds (sn -FanMorphW) . x in K0 let x, K0 be set ; ::_thesis: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies (sn -FanMorphW) . x in K0 ) assume A1: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: (sn -FanMorphW) . x in K0 then ex p being Point of (TOP-REAL 2) st ( p = x & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) ; hence (sn -FanMorphW) . x in K0 by A1, Th16; ::_thesis: verum end; scheme :: JGRAPH_4:sch 1 InclSub{ F1() -> non empty Subset of (TOP-REAL 2), P1[ set ] } : { p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | F1()) provided A1: F1() = NonZero (TOP-REAL 2) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) ) } or x in the carrier of ((TOP-REAL 2) | F1()) ) assume x in { p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in the carrier of ((TOP-REAL 2) | F1()) then A2: ex p being Point of (TOP-REAL 2) st ( x = p & P1[p] & p <> 0. (TOP-REAL 2) ) ; A3: F1() ` = {(0. (TOP-REAL 2))} by A1, JGRAPH_3:20; now__::_thesis:_x_in_F1() assume not x in F1() ; ::_thesis: contradiction then x in the carrier of (TOP-REAL 2) \ F1() by A2, XBOOLE_0:def_5; then x in F1() ` by SUBSET_1:def_4; hence contradiction by A3, A2, TARSKI:def_1; ::_thesis: verum end; hence x in the carrier of ((TOP-REAL 2) | F1()) by PRE_TOPC:8; ::_thesis: verum end; theorem Th36: :: JGRAPH_4:36 for sn being Real for D being non empty Subset of (TOP-REAL 2) st - 1 < sn & sn < 1 & D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphW) | D & h is continuous ) proof ( |[0,1]| `1 = 0 & |[0,1]| `2 = 1 ) by EUCLID:52; then A1: |[0,1]| in { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } by JGRAPH_2:3; set Y1 = |[0,1]|; defpred S1[ Point of (TOP-REAL 2)] means $1 `1 <= 0 ; reconsider B0 = {(0. (TOP-REAL 2))} as Subset of (TOP-REAL 2) ; let sn be Real; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st - 1 < sn & sn < 1 & D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphW) | D & h is continuous ) let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 & D ` = {(0. (TOP-REAL 2))} implies ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphW) | D & h is continuous ) ) assume that A2: ( - 1 < sn & sn < 1 ) and A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphW) | D & h is continuous ) A4: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; A5: D = B0 ` by A3 .= NonZero (TOP-REAL 2) by SUBSET_1:def_4 ; { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) from JGRAPH_4:sch_1(A5); then reconsider K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A1; A6: K0 = the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; A7: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; A8: rng ((sn -FanMorphW) | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((sn -FanMorphW) | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) ) assume y in rng ((sn -FanMorphW) | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0) then consider x being set such that A9: x in dom ((sn -FanMorphW) | K0) and A10: y = ((sn -FanMorphW) | K0) . x by FUNCT_1:def_3; x in (dom (sn -FanMorphW)) /\ K0 by A9, RELAT_1:61; then A11: x in K0 by XBOOLE_0:def_4; K0 c= the carrier of (TOP-REAL 2) by A7, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A11; (sn -FanMorphW) . p = y by A10, A11, FUNCT_1:49; then y in K0 by A2, A11, Th34; hence y in the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; ::_thesis: verum end; A12: K0 c= the carrier of (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K0 or z in the carrier of (TOP-REAL 2) ) assume z in K0 ; ::_thesis: z in the carrier of (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & p8 `1 <= 0 & p8 <> 0. (TOP-REAL 2) ) ; hence z in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; ( |[0,1]| `1 = 0 & |[0,1]| `2 = 1 ) by EUCLID:52; then A13: |[0,1]| in { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } by JGRAPH_2:3; A14: the carrier of ((TOP-REAL 2) | D) = NonZero (TOP-REAL 2) by A5, PRE_TOPC:8; defpred S2[ Point of (TOP-REAL 2)] means $1 `1 >= 0 ; { p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) from JGRAPH_4:sch_1(A5); then reconsider K1 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A13; A15: ( K0 is closed & K1 is closed ) by A5, Th29, Th31; dom ((sn -FanMorphW) | K0) = (dom (sn -FanMorphW)) /\ K0 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K0 by FUNCT_2:def_1 .= K0 by A12, XBOOLE_1:28 ; then reconsider f = (sn -FanMorphW) | K0 as Function of (((TOP-REAL 2) | D) | K0),((TOP-REAL 2) | D) by A6, A8, FUNCT_2:2, XBOOLE_1:1; A16: K1 = the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; A17: rng ((sn -FanMorphW) | K1) c= the carrier of (((TOP-REAL 2) | D) | K1) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((sn -FanMorphW) | K1) or y in the carrier of (((TOP-REAL 2) | D) | K1) ) assume y in rng ((sn -FanMorphW) | K1) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K1) then consider x being set such that A18: x in dom ((sn -FanMorphW) | K1) and A19: y = ((sn -FanMorphW) | K1) . x by FUNCT_1:def_3; x in (dom (sn -FanMorphW)) /\ K1 by A18, RELAT_1:61; then A20: x in K1 by XBOOLE_0:def_4; K1 c= the carrier of (TOP-REAL 2) by A7, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A20; (sn -FanMorphW) . p = y by A19, A20, FUNCT_1:49; then y in K1 by A2, A20, Th35; hence y in the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; ::_thesis: verum end; A21: K1 c= the carrier of (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in the carrier of (TOP-REAL 2) ) assume z in K1 ; ::_thesis: z in the carrier of (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & p8 `1 >= 0 & p8 <> 0. (TOP-REAL 2) ) ; hence z in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; dom ((sn -FanMorphW) | K1) = (dom (sn -FanMorphW)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by A21, XBOOLE_1:28 ; then reconsider g = (sn -FanMorphW) | K1 as Function of (((TOP-REAL 2) | D) | K1),((TOP-REAL 2) | D) by A16, A17, FUNCT_2:2, XBOOLE_1:1; A22: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5; A23: D c= K0 \/ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in K0 \/ K1 ) assume A24: x in D ; ::_thesis: x in K0 \/ K1 then reconsider px = x as Point of (TOP-REAL 2) ; not x in {(0. (TOP-REAL 2))} by A5, A24, XBOOLE_0:def_5; then ( ( px `1 <= 0 & px <> 0. (TOP-REAL 2) ) or ( px `1 >= 0 & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1; then ( x in K0 or x in K1 ) ; hence x in K0 \/ K1 by XBOOLE_0:def_3; ::_thesis: verum end; A25: dom f = K0 by A6, FUNCT_2:def_1; A26: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5; A27: for x being set st x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) holds f . x = g . x proof let x be set ; ::_thesis: ( x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) implies f . x = g . x ) assume A28: x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) ; ::_thesis: f . x = g . x then x in K0 by A26, XBOOLE_0:def_4; then f . x = (sn -FanMorphW) . x by FUNCT_1:49; hence f . x = g . x by A22, A28, FUNCT_1:49; ::_thesis: verum end; D = [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; then A29: ([#] (((TOP-REAL 2) | D) | K0)) \/ ([#] (((TOP-REAL 2) | D) | K1)) = [#] ((TOP-REAL 2) | D) by A26, A22, A23, XBOOLE_0:def_10; A30: ( f is continuous & g is continuous ) by A2, A5, Th30, Th32; then consider h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) such that A31: h = f +* g and h is continuous by A26, A22, A29, A15, A27, JGRAPH_2:1; A32: dom h = the carrier of ((TOP-REAL 2) | D) by FUNCT_2:def_1; A33: dom g = K1 by A16, FUNCT_2:def_1; ( K0 = [#] (((TOP-REAL 2) | D) | K0) & K1 = [#] (((TOP-REAL 2) | D) | K1) ) by PRE_TOPC:def_5; then A34: f tolerates g by A27, A25, A33, PARTFUN1:def_4; A35: for x being set st x in dom h holds h . x = ((sn -FanMorphW) | D) . x proof let x be set ; ::_thesis: ( x in dom h implies h . x = ((sn -FanMorphW) | D) . x ) assume A36: x in dom h ; ::_thesis: h . x = ((sn -FanMorphW) | D) . x then reconsider p = x as Point of (TOP-REAL 2) by A14, XBOOLE_0:def_5; A37: x in (D `) ` by A32, A36, PRE_TOPC:8; not x in {(0. (TOP-REAL 2))} by A14, A36, XBOOLE_0:def_5; then A38: x <> 0. (TOP-REAL 2) by TARSKI:def_1; percases ( x in K0 or not x in K0 ) ; supposeA39: x in K0 ; ::_thesis: h . x = ((sn -FanMorphW) | D) . x A40: ((sn -FanMorphW) | D) . p = (sn -FanMorphW) . p by A37, FUNCT_1:49 .= f . p by A39, FUNCT_1:49 ; h . p = (g +* f) . p by A31, A34, FUNCT_4:34 .= f . p by A25, A39, FUNCT_4:13 ; hence h . x = ((sn -FanMorphW) | D) . x by A40; ::_thesis: verum end; suppose not x in K0 ; ::_thesis: h . x = ((sn -FanMorphW) | D) . x then not p `1 <= 0 by A38; then A41: x in K1 by A38; ((sn -FanMorphW) | D) . p = (sn -FanMorphW) . p by A37, FUNCT_1:49 .= g . p by A41, FUNCT_1:49 ; hence h . x = ((sn -FanMorphW) | D) . x by A31, A33, A41, FUNCT_4:13; ::_thesis: verum end; end; end; dom (sn -FanMorphW) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then dom ((sn -FanMorphW) | D) = the carrier of (TOP-REAL 2) /\ D by RELAT_1:61 .= the carrier of ((TOP-REAL 2) | D) by A4, XBOOLE_1:28 ; then f +* g = (sn -FanMorphW) | D by A31, A32, A35, FUNCT_1:2; hence ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphW) | D & h is continuous ) by A26, A22, A29, A30, A15, A27, JGRAPH_2:1; ::_thesis: verum end; Lm11: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; theorem Th37: :: JGRAPH_4:37 for sn being Real st - 1 < sn & sn < 1 holds ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphW & h is continuous ) proof reconsider D = NonZero (TOP-REAL 2) as non empty Subset of (TOP-REAL 2) by JGRAPH_2:9; let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphW & h is continuous ) ) assume that A1: - 1 < sn and A2: sn < 1 ; ::_thesis: ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphW & h is continuous ) reconsider f = sn -FanMorphW as Function of (TOP-REAL 2),(TOP-REAL 2) ; A3: f . (0. (TOP-REAL 2)) = 0. (TOP-REAL 2) by Th16, JGRAPH_2:3; A4: for p being Point of ((TOP-REAL 2) | D) holds f . p <> f . (0. (TOP-REAL 2)) proof let p be Point of ((TOP-REAL 2) | D); ::_thesis: f . p <> f . (0. (TOP-REAL 2)) A5: [#] ((TOP-REAL 2) | D) = D by PRE_TOPC:def_5; then reconsider q = p as Point of (TOP-REAL 2) by XBOOLE_0:def_5; not p in {(0. (TOP-REAL 2))} by A5, XBOOLE_0:def_5; then A6: p <> 0. (TOP-REAL 2) by TARSKI:def_1; percases ( ( (q `2) / |.q.| >= sn & q `1 <= 0 ) or ( (q `2) / |.q.| < sn & q `1 <= 0 ) or q `1 > 0 ) ; supposeA7: ( (q `2) / |.q.| >= sn & q `1 <= 0 ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) set q9 = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]|; A8: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)) by EUCLID:52; A9: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; now__::_thesis:_not_|[(|.q.|_*_(-_(sqrt_(1_-_(((((q_`2)_/_|.q.|)_-_sn)_/_(1_-_sn))_^2))))),(|.q.|_*_((((q_`2)_/_|.q.|)_-_sn)_/_(1_-_sn)))]|_=_0._(TOP-REAL_2) assume A10: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction A11: |.q.| <> 0 ^2 by A6, TOPRNS_1:24; then - (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) = - (sqrt (1 - 0)) by A8, A10, JGRAPH_2:3, XCMPLX_1:6 .= - 1 by SQUARE_1:18 ; hence contradiction by A9, A10, A11, JGRAPH_2:3, XCMPLX_1:6; ::_thesis: verum end; hence f . p <> f . (0. (TOP-REAL 2)) by A1, A2, A3, A6, A7, Th18; ::_thesis: verum end; supposeA12: ( (q `2) / |.q.| < sn & q `1 <= 0 ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) set q9 = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|; A13: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) by EUCLID:52; A14: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))) by EUCLID:52; now__::_thesis:_not_|[(|.q.|_*_(-_(sqrt_(1_-_(((((q_`2)_/_|.q.|)_-_sn)_/_(1_+_sn))_^2))))),(|.q.|_*_((((q_`2)_/_|.q.|)_-_sn)_/_(1_+_sn)))]|_=_0._(TOP-REAL_2) assume A15: |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction A16: |.q.| <> 0 ^2 by A6, TOPRNS_1:24; then - (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) = - (sqrt (1 - 0)) by A13, A15, JGRAPH_2:3, XCMPLX_1:6 .= - 1 by SQUARE_1:18 ; hence contradiction by A14, A15, A16, JGRAPH_2:3, XCMPLX_1:6; ::_thesis: verum end; hence f . p <> f . (0. (TOP-REAL 2)) by A1, A2, A3, A6, A12, Th18; ::_thesis: verum end; suppose q `1 > 0 ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) then f . p = p by Th16; hence f . p <> f . (0. (TOP-REAL 2)) by A6, Th16, JGRAPH_2:3; ::_thesis: verum end; end; end; A17: for V being Subset of (TOP-REAL 2) st f . (0. (TOP-REAL 2)) in V & V is open holds ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) proof reconsider u0 = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:67; let V be Subset of (TOP-REAL 2); ::_thesis: ( f . (0. (TOP-REAL 2)) in V & V is open implies ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) ) reconsider VV = V as Subset of (TopSpaceMetr (Euclid 2)) by Lm11; assume that A18: f . (0. (TOP-REAL 2)) in V and A19: V is open ; ::_thesis: ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) VV is open by A19, Lm11, PRE_TOPC:30; then consider r being real number such that A20: r > 0 and A21: Ball (u0,r) c= V by A3, A18, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider W1 = Ball (u0,r) as Subset of (TOP-REAL 2) ; A22: W1 is open by GOBOARD6:3; A23: f .: W1 c= W1 proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f .: W1 or z in W1 ) assume z in f .: W1 ; ::_thesis: z in W1 then consider y being set such that A24: y in dom f and A25: y in W1 and A26: z = f . y by FUNCT_1:def_6; z in rng f by A24, A26, FUNCT_1:def_3; then reconsider qz = z as Point of (TOP-REAL 2) ; reconsider pz = qz as Point of (Euclid 2) by EUCLID:67; reconsider q = y as Point of (TOP-REAL 2) by A24; reconsider qy = q as Point of (Euclid 2) by EUCLID:67; dist (u0,qy) < r by A25, METRIC_1:11; then A27: |.((0. (TOP-REAL 2)) - q).| < r by JGRAPH_1:28; percases ( q `1 >= 0 or ( q <> 0. (TOP-REAL 2) & (q `2) / |.q.| >= sn & q `1 <= 0 ) or ( q <> 0. (TOP-REAL 2) & (q `2) / |.q.| < sn & q `1 <= 0 ) ) by JGRAPH_2:3; suppose q `1 >= 0 ; ::_thesis: z in W1 hence z in W1 by A25, A26, Th16; ::_thesis: verum end; supposeA28: ( q <> 0. (TOP-REAL 2) & (q `2) / |.q.| >= sn & q `1 <= 0 ) ; ::_thesis: z in W1 then A29: ((q `2) / |.q.|) - sn >= 0 by XREAL_1:48; 0 <= (q `1) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A30: ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; A31: 1 - sn > 0 by A2, XREAL_1:149; |.q.| <> 0 by A28, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A30, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `2) / |.q.| by SQUARE_1:51; then 1 - sn >= ((q `2) / |.q.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((q `2) / |.q.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A31, XREAL_1:72; then - 1 <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A31, XCMPLX_1:197; then ((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A31, A29, SQUARE_1:49; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A32: 1 - ((- ((((q `2) / |.q.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; A33: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, A28, Th18; then A34: qz `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)) by A26, EUCLID:52; qz `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))) by A26, A33, EUCLID:52; then A35: (qz `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) by A32, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A34, A35 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; then A36: |.qz.| = |.q.| by SQUARE_1:22; |.(- q).| < r by A27, EUCLID:27; then |.q.| < r by TOPRNS_1:26; then |.(- qz).| < r by A36, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - qz).| < r by EUCLID:27; then dist (u0,pz) < r by JGRAPH_1:28; hence z in W1 by METRIC_1:11; ::_thesis: verum end; supposeA37: ( q <> 0. (TOP-REAL 2) & (q `2) / |.q.| < sn & q `1 <= 0 ) ; ::_thesis: z in W1 0 <= (q `1) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A38: ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; A39: 1 + sn > 0 by A1, XREAL_1:148; |.q.| <> 0 by A37, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A38, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `2) / |.q.| by SQUARE_1:51; then - (- 1) >= - ((q `2) / |.q.|) by XREAL_1:24; then 1 + sn >= (- ((q `2) / |.q.|)) + sn by XREAL_1:7; then A40: (- (((q `2) / |.q.|) - sn)) / (1 + sn) <= 1 by A39, XREAL_1:185; sn - ((q `2) / |.q.|) >= 0 by A37, XREAL_1:48; then - 1 <= (- (((q `2) / |.q.|) - sn)) / (1 + sn) by A39; then ((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A40, SQUARE_1:49; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then A41: 1 - ((- ((((q `2) / |.q.|) - sn) / (1 + sn))) ^2) >= 0 by XCMPLX_1:187; A42: (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A1, A2, A37, Th18; then A43: qz `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) by A26, EUCLID:52; qz `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))) by A26, A42, EUCLID:52; then A44: (qz `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) by A41, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A43, A44 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; then A45: |.qz.| = |.q.| by SQUARE_1:22; |.(- q).| < r by A27, EUCLID:27; then |.q.| < r by TOPRNS_1:26; then |.(- qz).| < r by A45, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - qz).| < r by EUCLID:27; then dist (u0,pz) < r by JGRAPH_1:28; hence z in W1 by METRIC_1:11; ::_thesis: verum end; end; end; u0 in W1 by A20, GOBOARD6:1; hence ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) by A21, A22, A23, XBOOLE_1:1; ::_thesis: verum end; A46: D ` = {(0. (TOP-REAL 2))} by JGRAPH_3:20; then ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphW) | D & h is continuous ) by A1, A2, Th36; hence ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphW & h is continuous ) by A3, A46, A4, A17, JGRAPH_3:3; ::_thesis: verum end; theorem Th38: :: JGRAPH_4:38 for sn being Real st - 1 < sn & sn < 1 holds sn -FanMorphW is one-to-one proof let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies sn -FanMorphW is one-to-one ) assume that A1: - 1 < sn and A2: sn < 1 ; ::_thesis: sn -FanMorphW is one-to-one for x1, x2 being set st x1 in dom (sn -FanMorphW) & x2 in dom (sn -FanMorphW) & (sn -FanMorphW) . x1 = (sn -FanMorphW) . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom (sn -FanMorphW) & x2 in dom (sn -FanMorphW) & (sn -FanMorphW) . x1 = (sn -FanMorphW) . x2 implies x1 = x2 ) assume that A3: x1 in dom (sn -FanMorphW) and A4: x2 in dom (sn -FanMorphW) and A5: (sn -FanMorphW) . x1 = (sn -FanMorphW) . x2 ; ::_thesis: x1 = x2 reconsider p2 = x2 as Point of (TOP-REAL 2) by A4; reconsider p1 = x1 as Point of (TOP-REAL 2) by A3; set q = p1; set p = p2; A6: 1 - sn > 0 by A2, XREAL_1:149; now__::_thesis:_(_(_p1_`1_>=_0_&_x1_=_x2_)_or_(_(p1_`2)_/_|.p1.|_>=_sn_&_p1_`1_<=_0_&_p1_<>_0._(TOP-REAL_2)_&_x1_=_x2_)_or_(_(p1_`2)_/_|.p1.|_<_sn_&_p1_`1_<=_0_&_p1_<>_0._(TOP-REAL_2)_&_x1_=_x2_)_) percases ( p1 `1 >= 0 or ( (p1 `2) / |.p1.| >= sn & p1 `1 <= 0 & p1 <> 0. (TOP-REAL 2) ) or ( (p1 `2) / |.p1.| < sn & p1 `1 <= 0 & p1 <> 0. (TOP-REAL 2) ) ) by JGRAPH_2:3; caseA7: p1 `1 >= 0 ; ::_thesis: x1 = x2 then A8: (sn -FanMorphW) . p1 = p1 by Th16; now__::_thesis:_(_(_p2_`1_>=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_>=_sn_&_p2_`1_<=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_<_sn_&_p2_`1_<=_0_&_x1_=_x2_)_) percases ( p2 `1 >= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 <= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 <= 0 ) ) by JGRAPH_2:3; case p2 `1 >= 0 ; ::_thesis: x1 = x2 hence x1 = x2 by A5, A8, Th16; ::_thesis: verum end; caseA9: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 <= 0 ) ; ::_thesis: x1 = x2 set p4 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|; A10: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; 0 <= (p2 `1) ^2 by XREAL_1:63; then 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) by XREAL_1:7; then A11: ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by A10, XREAL_1:72; A12: |.p2.| > 0 by A9, Lm1; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A11, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p2 `2) / |.p2.| by SQUARE_1:51; then 1 - sn >= ((p2 `2) / |.p2.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((p2 `2) / |.p2.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((p2 `2) / |.p2.|) - sn)) / (1 - sn) by A6, XREAL_1:72; then A13: - 1 <= (- (((p2 `2) / |.p2.|) - sn)) / (1 - sn) by A6, XCMPLX_1:197; A14: ((p2 `2) / |.p2.|) - sn >= 0 by A9, XREAL_1:48; A15: (sn -FanMorphW) . p2 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| by A1, A2, A9, Th18; ((p2 `2) / |.p2.|) - sn >= 0 by A9, XREAL_1:48; then ((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A6, A13, SQUARE_1:49; then A16: 1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p2 `2) / |.p2.|) - sn)) ^2) / ((1 - sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p2 `2) / |.p2.|) - sn) ^2) / ((1 - sn) ^2))) >= 0 ; then sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)) >= 0 by XCMPLX_1:76; then ( |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `1 = |.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))) & p1 `1 = 0 ) by A5, A7, A8, A15, EUCLID:52; then A17: - (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))) = 0 by A5, A8, A15, A12, XCMPLX_1:6; 1 - ((- ((((p2 `2) / |.p2.|) - sn) / (1 - sn))) ^2) >= 0 by A16, XCMPLX_1:187; then 1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2) = 0 by A17, SQUARE_1:24; then 1 = (((p2 `2) / |.p2.|) - sn) / (1 - sn) by A6, A14, SQUARE_1:18, SQUARE_1:22; then 1 * (1 - sn) = ((p2 `2) / |.p2.|) - sn by A6, XCMPLX_1:87; then 1 * |.p2.| = p2 `2 by A12, XCMPLX_1:87; then p2 `1 = 0 by A10, XCMPLX_1:6; hence x1 = x2 by A5, A8, Th16; ::_thesis: verum end; caseA18: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 <= 0 ) ; ::_thesis: x1 = x2 then A19: (sn -FanMorphW) . p2 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| by A1, A2, Th18; set p4 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|; A20: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; 0 <= (p2 `1) ^2 by XREAL_1:63; then 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) by XREAL_1:7; then A21: ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by A20, XREAL_1:72; A22: 1 + sn > 0 by A1, XREAL_1:148; A23: ((p2 `2) / |.p2.|) - sn <= 0 by A18, XREAL_1:47; then A24: - 1 <= (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) by A22; A25: |.p2.| > 0 by A18, Lm1; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A21, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (- ((p2 `2) / |.p2.|)) ^2 <= 1 ; then 1 >= - ((p2 `2) / |.p2.|) by SQUARE_1:51; then 1 + sn >= (- ((p2 `2) / |.p2.|)) + sn by XREAL_1:7; then (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) <= 1 by A22, XREAL_1:185; then ((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A24, SQUARE_1:49; then A26: 1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p2 `2) / |.p2.|) - sn)) ^2) / ((1 + sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p2 `2) / |.p2.|) - sn) ^2) / ((1 + sn) ^2))) >= 0 ; then sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)) >= 0 by XCMPLX_1:76; then ( |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `1 = |.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))) & p1 `1 = 0 ) by A5, A7, A8, A19, EUCLID:52; then A27: - (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))) = 0 by A5, A8, A19, A25, XCMPLX_1:6; 1 - ((- ((((p2 `2) / |.p2.|) - sn) / (1 + sn))) ^2) >= 0 by A26, XCMPLX_1:187; then 1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2) = 0 by A27, SQUARE_1:24; then 1 = sqrt ((- ((((p2 `2) / |.p2.|) - sn) / (1 + sn))) ^2) by SQUARE_1:18; then 1 = - ((((p2 `2) / |.p2.|) - sn) / (1 + sn)) by A22, A23, SQUARE_1:22; then 1 = (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) by XCMPLX_1:187; then 1 * (1 + sn) = - (((p2 `2) / |.p2.|) - sn) by A22, XCMPLX_1:87; then (1 + sn) - sn = - ((p2 `2) / |.p2.|) ; then 1 = (- (p2 `2)) / |.p2.| by XCMPLX_1:187; then 1 * |.p2.| = - (p2 `2) by A25, XCMPLX_1:87; then ((p2 `2) ^2) - ((p2 `2) ^2) = (p2 `1) ^2 by A20, XCMPLX_1:26; then p2 `1 = 0 by XCMPLX_1:6; hence x1 = x2 by A5, A8, Th16; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; caseA28: ( (p1 `2) / |.p1.| >= sn & p1 `1 <= 0 & p1 <> 0. (TOP-REAL 2) ) ; ::_thesis: x1 = x2 then |.p1.| > 0 by Lm1; then A29: |.p1.| ^2 > 0 by SQUARE_1:12; set q4 = |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]|; A30: |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `2 = |.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)) by EUCLID:52; A31: (sn -FanMorphW) . p1 = |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| by A1, A2, A28, Th18; A32: |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `1 = |.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; now__::_thesis:_(_(_p2_`1_>=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_>=_sn_&_p2_`1_<=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_<_sn_&_p2_`1_<=_0_&_x1_=_x2_)_) percases ( p2 `1 >= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 <= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 <= 0 ) ) by JGRAPH_2:3; caseA33: p2 `1 >= 0 ; ::_thesis: x1 = x2 A34: ((p1 `2) / |.p1.|) - sn >= 0 by A28, XREAL_1:48; A35: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1; 0 <= (p1 `1) ^2 by XREAL_1:63; then 0 + ((p1 `2) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) by XREAL_1:7; then A36: ((p1 `2) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by A35, XREAL_1:72; A37: (sn -FanMorphW) . p2 = p2 by A33, Th16; A38: ((p1 `2) / |.p1.|) - sn >= 0 by A28, XREAL_1:48; A39: 1 - sn > 0 by A2, XREAL_1:149; A40: |.p1.| > 0 by A28, Lm1; then |.p1.| ^2 > 0 by SQUARE_1:12; then ((p1 `2) ^2) / (|.p1.| ^2) <= 1 by A36, XCMPLX_1:60; then ((p1 `2) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p1 `2) / |.p1.| by SQUARE_1:51; then 1 - sn >= ((p1 `2) / |.p1.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((p1 `2) / |.p1.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((p1 `2) / |.p1.|) - sn)) / (1 - sn) by A39, XREAL_1:72; then - 1 <= (- (((p1 `2) / |.p1.|) - sn)) / (1 - sn) by A39, XCMPLX_1:197; then ((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A39, A34, SQUARE_1:49; then A41: 1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p1 `2) / |.p1.|) - sn)) ^2) / ((1 - sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) ^2) / ((1 - sn) ^2))) >= 0 ; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)) >= 0 by XCMPLX_1:76; then p2 `1 = 0 by A5, A31, A33, A37, EUCLID:52; then A42: - (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))) = 0 by A5, A31, A32, A37, A40, XCMPLX_1:6; 1 - ((- ((((p1 `2) / |.p1.|) - sn) / (1 - sn))) ^2) >= 0 by A41, XCMPLX_1:187; then 1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2) = 0 by A42, SQUARE_1:24; then 1 = (((p1 `2) / |.p1.|) - sn) / (1 - sn) by A39, A38, SQUARE_1:18, SQUARE_1:22; then 1 * (1 - sn) = ((p1 `2) / |.p1.|) - sn by A39, XCMPLX_1:87; then 1 * |.p1.| = p1 `2 by A40, XCMPLX_1:87; then p1 `1 = 0 by A35, XCMPLX_1:6; hence x1 = x2 by A5, A37, Th16; ::_thesis: verum end; caseA43: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 <= 0 ) ; ::_thesis: x1 = x2 0 <= (p1 `1) ^2 by XREAL_1:63; then ( |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) & 0 + ((p1 `2) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p1 `2) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by XREAL_1:72; then ((p1 `2) ^2) / (|.p1.| ^2) <= 1 by A29, XCMPLX_1:60; then ((p1 `2) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p1 `2) / |.p1.| by SQUARE_1:51; then 1 - sn >= ((p1 `2) / |.p1.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((p1 `2) / |.p1.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((p1 `2) / |.p1.|) - sn)) / (1 - sn) by A6, XREAL_1:72; then A44: - 1 <= (- (((p1 `2) / |.p1.|) - sn)) / (1 - sn) by A6, XCMPLX_1:197; ((p1 `2) / |.p1.|) - sn >= 0 by A28, XREAL_1:48; then ((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A6, A44, SQUARE_1:49; then 1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A45: 1 - ((- ((((p1 `2) / |.p1.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `1 = |.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; then A46: (|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `1) ^2 = (|.p1.| ^2) * ((sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))) ^2) .= (|.p1.| ^2) * (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)) by A45, SQUARE_1:def_2 ; A47: |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `2 = |.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)) by EUCLID:52; |.|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]|.| ^2 = ((|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `1) ^2) + ((|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `2) ^2) by JGRAPH_3:1 .= |.p1.| ^2 by A47, A46 ; then sqrt (|.|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]|.| ^2) = |.p1.| by SQUARE_1:22; then A48: |.|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]|.| = |.p1.| by SQUARE_1:22; 0 <= (p2 `1) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A49: ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; A50: |.p2.| > 0 by A43, Lm1; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A49, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p2 `2) / |.p2.| by SQUARE_1:51; then 1 - sn >= ((p2 `2) / |.p2.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((p2 `2) / |.p2.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((p2 `2) / |.p2.|) - sn)) / (1 - sn) by A6, XREAL_1:72; then A51: - 1 <= (- (((p2 `2) / |.p2.|) - sn)) / (1 - sn) by A6, XCMPLX_1:197; set p4 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|; A52: |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)) by EUCLID:52; ((p2 `2) / |.p2.|) - sn >= 0 by A43, XREAL_1:48; then ((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A6, A51, SQUARE_1:49; then 1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A53: 1 - ((- ((((p2 `2) / |.p2.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `1 = |.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; then A54: (|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `1) ^2 = (|.p2.| ^2) * ((sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))) ^2) .= (|.p2.| ^2) * (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)) by A53, SQUARE_1:def_2 ; |.|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|.| ^2 = ((|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `1) ^2) + ((|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `2) ^2) by JGRAPH_3:1 .= |.p2.| ^2 by A52, A54 ; then sqrt (|.|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|.| ^2) = |.p2.| by SQUARE_1:22; then A55: |.|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|.| = |.p2.| by SQUARE_1:22; A56: (sn -FanMorphW) . p2 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| by A1, A2, A43, Th18; then (((p2 `2) / |.p2.|) - sn) / (1 - sn) = (|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn))) / |.p2.| by A5, A31, A30, A52, A50, XCMPLX_1:89; then (((p2 `2) / |.p2.|) - sn) / (1 - sn) = (((p1 `2) / |.p1.|) - sn) / (1 - sn) by A5, A31, A56, A48, A50, A55, XCMPLX_1:89; then ((((p2 `2) / |.p2.|) - sn) / (1 - sn)) * (1 - sn) = ((p1 `2) / |.p1.|) - sn by A6, XCMPLX_1:87; then ((p2 `2) / |.p2.|) - sn = ((p1 `2) / |.p1.|) - sn by A6, XCMPLX_1:87; then ((p2 `2) / |.p2.|) * |.p2.| = p1 `2 by A5, A31, A56, A48, A50, A55, XCMPLX_1:87; then A57: p2 `2 = p1 `2 by A50, XCMPLX_1:87; ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1; then (- (p2 `1)) ^2 = (p1 `1) ^2 by A5, A31, A56, A48, A55, A57; then - (p2 `1) = sqrt ((- (p1 `1)) ^2) by A43, SQUARE_1:22; then A58: - (- (p2 `1)) = - (- (p1 `1)) by A28, SQUARE_1:22; p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53; hence x1 = x2 by A57, A58, EUCLID:53; ::_thesis: verum end; caseA59: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 <= 0 ) ; ::_thesis: x1 = x2 then ((p2 `2) / |.p2.|) - sn < 0 by XREAL_1:49; then A60: (((p2 `2) / |.p2.|) - sn) / (1 + sn) < 0 by A1, XREAL_1:141, XREAL_1:148; set p4 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|; A61: ( |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)) & ((p1 `2) / |.p1.|) - sn >= 0 ) by A28, EUCLID:52, XREAL_1:48; A62: 1 - sn > 0 by A2, XREAL_1:149; (sn -FanMorphW) . p2 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| by A1, A2, A59, Th18; hence x1 = x2 by A5, A31, A30, A59, A60, A61, A62, Lm1, XREAL_1:132; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; caseA63: ( (p1 `2) / |.p1.| < sn & p1 `1 <= 0 & p1 <> 0. (TOP-REAL 2) ) ; ::_thesis: verum then A64: |.p1.| > 0 by Lm1; then A65: |.p1.| ^2 > 0 by SQUARE_1:12; set q4 = |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]|; A66: |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `1 = |.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))) by EUCLID:52; A67: |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `2 = |.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)) by EUCLID:52; A68: (sn -FanMorphW) . p1 = |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| by A1, A2, A63, Th18; percases ( p2 `1 >= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 <= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 <= 0 ) ) by JGRAPH_2:3; supposeA69: p2 `1 >= 0 ; ::_thesis: x1 = x2 A70: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1; A71: 1 + sn > 0 by A1, XREAL_1:148; 0 <= (p1 `1) ^2 by XREAL_1:63; then 0 + ((p1 `2) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) by XREAL_1:7; then ((p1 `2) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by A70, XREAL_1:72; then ((p1 `2) ^2) / (|.p1.| ^2) <= 1 by A65, XCMPLX_1:60; then ((p1 `2) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then (- ((p1 `2) / |.p1.|)) ^2 <= 1 ; then 1 >= - ((p1 `2) / |.p1.|) by SQUARE_1:51; then 1 + sn >= (- ((p1 `2) / |.p1.|)) + sn by XREAL_1:7; then A72: (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) <= 1 by A71, XREAL_1:185; A73: ((p1 `2) / |.p1.|) - sn <= 0 by A63, XREAL_1:47; then - 1 <= (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) by A71; then ((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A72, SQUARE_1:49; then A74: 1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then A75: 1 - ((- ((((p1 `2) / |.p1.|) - sn) / (1 + sn))) ^2) >= 0 by XCMPLX_1:187; A76: (sn -FanMorphW) . p2 = p2 by A69, Th16; sqrt (1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2)) >= 0 by A74, SQUARE_1:def_2; then sqrt (1 - (((- (((p1 `2) / |.p1.|) - sn)) ^2) / ((1 + sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) ^2) / ((1 + sn) ^2))) >= 0 ; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)) >= 0 by XCMPLX_1:76; then p2 `1 = 0 by A5, A68, A69, A76, EUCLID:52; then - (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))) = 0 by A5, A68, A66, A64, A76, XCMPLX_1:6; then 1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2) = 0 by A75, SQUARE_1:24; then 1 = sqrt ((- ((((p1 `2) / |.p1.|) - sn) / (1 + sn))) ^2) by SQUARE_1:18; then 1 = - ((((p1 `2) / |.p1.|) - sn) / (1 + sn)) by A71, A73, SQUARE_1:22; then 1 = (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) by XCMPLX_1:187; then 1 * (1 + sn) = - (((p1 `2) / |.p1.|) - sn) by A71, XCMPLX_1:87; then (1 + sn) - sn = - ((p1 `2) / |.p1.|) ; then 1 = (- (p1 `2)) / |.p1.| by XCMPLX_1:187; then 1 * |.p1.| = - (p1 `2) by A64, XCMPLX_1:87; then ((p1 `2) ^2) - ((p1 `2) ^2) = (p1 `1) ^2 by A70, XCMPLX_1:26; then p1 `1 = 0 by XCMPLX_1:6; hence x1 = x2 by A5, A76, Th16; ::_thesis: verum end; supposeA77: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 <= 0 ) ; ::_thesis: x1 = x2 set p4 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|; A78: ( |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)) & 1 - sn > 0 ) by A2, EUCLID:52, XREAL_1:149; ((p1 `2) / |.p1.|) - sn < 0 by A63, XREAL_1:49; then A79: (((p1 `2) / |.p1.|) - sn) / (1 + sn) < 0 by A1, XREAL_1:141, XREAL_1:148; ( (sn -FanMorphW) . p2 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| & ((p2 `2) / |.p2.|) - sn >= 0 ) by A1, A2, A77, Th18, XREAL_1:48; hence x1 = x2 by A5, A63, A68, A67, A79, A78, Lm1, XREAL_1:132; ::_thesis: verum end; supposeA80: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 <= 0 ) ; ::_thesis: x1 = x2 0 <= (p2 `1) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A81: ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; A82: 1 + sn > 0 by A1, XREAL_1:148; 0 <= (p1 `1) ^2 by XREAL_1:63; then ( |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) & 0 + ((p1 `2) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p1 `2) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by XREAL_1:72; then ((p1 `2) ^2) / (|.p1.| ^2) <= 1 by A65, XCMPLX_1:60; then ((p1 `2) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p1 `2) / |.p1.| by SQUARE_1:51; then (- 1) - sn <= ((p1 `2) / |.p1.|) - sn by XREAL_1:9; then - ((- 1) - sn) >= - (((p1 `2) / |.p1.|) - sn) by XREAL_1:24; then A83: (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) <= 1 by A82, XREAL_1:185; ((p1 `2) / |.p1.|) - sn <= 0 by A63, XREAL_1:47; then - 1 <= (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) by A82; then ((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A83, SQUARE_1:49; then 1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then A84: 1 - ((- ((((p1 `2) / |.p1.|) - sn) / (1 + sn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `1 = |.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))) by EUCLID:52; then A85: (|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `1) ^2 = (|.p1.| ^2) * ((sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))) ^2) .= (|.p1.| ^2) * (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)) by A84, SQUARE_1:def_2 ; A86: |[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `2 = |.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)) by EUCLID:52; set p4 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|; A87: |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)) by EUCLID:52; |.|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]|.| ^2 = ((|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `1) ^2) + ((|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `2) ^2) by JGRAPH_3:1 .= |.p1.| ^2 by A86, A85 ; then sqrt (|.|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]|.| ^2) = |.p1.| by SQUARE_1:22; then A88: |.|[(|.p1.| * (- (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]|.| = |.p1.| by SQUARE_1:22; ((p2 `2) / |.p2.|) - sn <= 0 by A80, XREAL_1:47; then A89: - 1 <= (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) by A82; A90: |.p2.| > 0 by A80, Lm1; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A81, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p2 `2) / |.p2.| by SQUARE_1:51; then (- 1) - sn <= ((p2 `2) / |.p2.|) - sn by XREAL_1:9; then - ((- 1) - sn) >= - (((p2 `2) / |.p2.|) - sn) by XREAL_1:24; then (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) <= 1 by A82, XREAL_1:185; then ((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A89, SQUARE_1:49; then 1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then A91: 1 - ((- ((((p2 `2) / |.p2.|) - sn) / (1 + sn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `1 = |.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))) by EUCLID:52; then A92: (|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `1) ^2 = (|.p2.| ^2) * ((sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))) ^2) .= (|.p2.| ^2) * (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)) by A91, SQUARE_1:def_2 ; |.|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|.| ^2 = ((|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `1) ^2) + ((|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `2) ^2) by JGRAPH_3:1 .= |.p2.| ^2 by A87, A92 ; then sqrt (|.|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|.| ^2) = |.p2.| by SQUARE_1:22; then A93: |.|[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|.| = |.p2.| by SQUARE_1:22; A94: (sn -FanMorphW) . p2 = |[(|.p2.| * (- (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| by A1, A2, A80, Th18; then (((p2 `2) / |.p2.|) - sn) / (1 + sn) = (|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn))) / |.p2.| by A5, A68, A67, A87, A90, XCMPLX_1:89; then (((p2 `2) / |.p2.|) - sn) / (1 + sn) = (((p1 `2) / |.p1.|) - sn) / (1 + sn) by A5, A68, A94, A88, A90, A93, XCMPLX_1:89; then ((((p2 `2) / |.p2.|) - sn) / (1 + sn)) * (1 + sn) = ((p1 `2) / |.p1.|) - sn by A82, XCMPLX_1:87; then ((p2 `2) / |.p2.|) - sn = ((p1 `2) / |.p1.|) - sn by A82, XCMPLX_1:87; then ((p2 `2) / |.p2.|) * |.p2.| = p1 `2 by A5, A68, A94, A88, A90, A93, XCMPLX_1:87; then A95: p2 `2 = p1 `2 by A90, XCMPLX_1:87; ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1; then (- (p2 `1)) ^2 = (p1 `1) ^2 by A5, A68, A94, A88, A93, A95; then - (p2 `1) = sqrt ((- (p1 `1)) ^2) by A80, SQUARE_1:22; then A96: - (- (p2 `1)) = - (- (p1 `1)) by A63, SQUARE_1:22; p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53; hence x1 = x2 by A95, A96, EUCLID:53; ::_thesis: verum end; end; end; end; end; hence x1 = x2 ; ::_thesis: verum end; hence sn -FanMorphW is one-to-one by FUNCT_1:def_4; ::_thesis: verum end; theorem Th39: :: JGRAPH_4:39 for sn being Real st - 1 < sn & sn < 1 holds ( sn -FanMorphW is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) ) proof let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies ( sn -FanMorphW is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) ) ) assume that A1: - 1 < sn and A2: sn < 1 ; ::_thesis: ( sn -FanMorphW is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) ) thus sn -FanMorphW is Function of (TOP-REAL 2),(TOP-REAL 2) ; ::_thesis: rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = sn -FanMorphW holds rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) proof let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = sn -FanMorphW implies rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) ) assume A3: f = sn -FanMorphW ; ::_thesis: rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) A4: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; the carrier of (TOP-REAL 2) c= rng f proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (TOP-REAL 2) or y in rng f ) assume y in the carrier of (TOP-REAL 2) ; ::_thesis: y in rng f then reconsider p2 = y as Point of (TOP-REAL 2) ; set q = p2; now__::_thesis:_(_(_p2_`1_>=_0_&_ex_x_being_set_st_ (_x_in_dom_(sn_-FanMorphW)_&_y_=_(sn_-FanMorphW)_._x_)_)_or_(_(p2_`2)_/_|.p2.|_>=_0_&_p2_`1_<=_0_&_p2_<>_0._(TOP-REAL_2)_&_ex_x_being_set_st_ (_x_in_dom_(sn_-FanMorphW)_&_y_=_(sn_-FanMorphW)_._x_)_)_or_(_(p2_`2)_/_|.p2.|_<_0_&_p2_`1_<=_0_&_p2_<>_0._(TOP-REAL_2)_&_ex_x_being_set_st_ (_x_in_dom_(sn_-FanMorphW)_&_y_=_(sn_-FanMorphW)_._x_)_)_) percases ( p2 `1 >= 0 or ( (p2 `2) / |.p2.| >= 0 & p2 `1 <= 0 & p2 <> 0. (TOP-REAL 2) ) or ( (p2 `2) / |.p2.| < 0 & p2 `1 <= 0 & p2 <> 0. (TOP-REAL 2) ) ) by JGRAPH_2:3; case p2 `1 >= 0 ; ::_thesis: ex x being set st ( x in dom (sn -FanMorphW) & y = (sn -FanMorphW) . x ) then y = (sn -FanMorphW) . p2 by Th16; hence ex x being set st ( x in dom (sn -FanMorphW) & y = (sn -FanMorphW) . x ) by A3, A4; ::_thesis: verum end; caseA5: ( (p2 `2) / |.p2.| >= 0 & p2 `1 <= 0 & p2 <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st ( x in dom (sn -FanMorphW) & y = (sn -FanMorphW) . x ) A6: - (- (1 + sn)) > 0 by A1, XREAL_1:148; A7: 1 - sn >= 0 by A2, XREAL_1:149; then ((p2 `2) / |.p2.|) * (1 - sn) >= 0 by A5; then - (1 + sn) <= ((p2 `2) / |.p2.|) * (1 - sn) by A6; then A8: ((- 1) - sn) + sn <= (((p2 `2) / |.p2.|) * (1 - sn)) + sn by XREAL_1:7; set px = |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|; A9: |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn) by EUCLID:52; |.p2.| <> 0 by A5, TOPRNS_1:24; then A10: |.p2.| ^2 > 0 by SQUARE_1:12; A11: |.p2.| > 0 by A5, Lm1; A12: dom (sn -FanMorphW) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A13: 1 - sn > 0 by A2, XREAL_1:149; 0 <= (p2 `1) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A10, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (p2 `2) / |.p2.| <= 1 by SQUARE_1:51; then ((p2 `2) / |.p2.|) * (1 - sn) <= 1 * (1 - sn) by A13, XREAL_1:64; then ((((p2 `2) / |.p2.|) * (1 - sn)) + sn) - sn <= 1 - sn ; then (((p2 `2) / |.p2.|) * (1 - sn)) + sn <= 1 by XREAL_1:9; then 1 ^2 >= ((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2 by A8, SQUARE_1:49; then A14: 1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2) >= 0 by XREAL_1:48; then A15: sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)) >= 0 by SQUARE_1:def_2; A16: |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `1 = - (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))) by EUCLID:52; then |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 = ((- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))) ^2) + ((|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn)) ^2) by A9, JGRAPH_3:1 .= ((|.p2.| ^2) * ((sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))) ^2)) + ((|.p2.| ^2) * (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)) ; then A17: |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 = ((|.p2.| ^2) * (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))) + ((|.p2.| ^2) * (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)) by A14, SQUARE_1:def_2 .= |.p2.| ^2 ; then A18: |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| = sqrt (|.p2.| ^2) by SQUARE_1:22 .= |.p2.| by SQUARE_1:22 ; then A19: |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| <> 0. (TOP-REAL 2) by A5, TOPRNS_1:23, TOPRNS_1:24; (((p2 `2) / |.p2.|) * (1 - sn)) + sn >= 0 + sn by A5, A7, XREAL_1:7; then (|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| >= sn by A5, A9, A18, TOPRNS_1:24, XCMPLX_1:89; then A20: (sn -FanMorphW) . |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| = |[(|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (1 - (((((|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) ^2))))),(|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * ((((|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)))]| by A1, A2, A16, A15, A19, Th18; A21: |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (((p2 `1) / |.p2.|) ^2))) = |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (- ((p2 `1) / |.p2.|))) by A5, SQUARE_1:23 .= p2 `1 by A11, A18, XCMPLX_1:87 ; A22: |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * ((((|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) = |.p2.| * ((((((p2 `2) / |.p2.|) * (1 - sn)) + sn) - sn) / (1 - sn)) by A5, A9, A18, TOPRNS_1:24, XCMPLX_1:89 .= |.p2.| * ((p2 `2) / |.p2.|) by A13, XCMPLX_1:89 .= p2 `2 by A5, TOPRNS_1:24, XCMPLX_1:87 ; then |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (1 - (((((|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) ^2)))) = |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (1 - (((p2 `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) ^2)))) by A5, A18, TOPRNS_1:24, XCMPLX_1:89 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (1 - (((p2 `2) ^2) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2))))) by XCMPLX_1:76 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (((|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2)) - (((p2 `2) ^2) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2))))) by A10, A17, XCMPLX_1:60 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (((|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2) - ((p2 `2) ^2)) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2)))) by XCMPLX_1:120 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (((((p2 `1) ^2) + ((p2 `2) ^2)) - ((p2 `2) ^2)) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2)))) by A17, JGRAPH_3:1 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (- (sqrt (((p2 `1) / |.p2.|) ^2))) by A18, XCMPLX_1:76 ; hence ex x being set st ( x in dom (sn -FanMorphW) & y = (sn -FanMorphW) . x ) by A20, A22, A21, A12, EUCLID:53; ::_thesis: verum end; caseA23: ( (p2 `2) / |.p2.| < 0 & p2 `1 <= 0 & p2 <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st ( x in dom (sn -FanMorphW) & y = (sn -FanMorphW) . x ) A24: 1 + sn >= 0 by A1, XREAL_1:148; then ((p2 `2) / |.p2.|) * (1 + sn) <= 0 by A23; then 1 - sn >= ((p2 `2) / |.p2.|) * (1 + sn) by A2, XREAL_1:149; then A25: (1 - sn) + sn >= (((p2 `2) / |.p2.|) * (1 + sn)) + sn by XREAL_1:7; set px = |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|; A26: |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn) by EUCLID:52; |.p2.| <> 0 by A23, TOPRNS_1:24; then A27: |.p2.| ^2 > 0 by SQUARE_1:12; A28: |.p2.| > 0 by A23, Lm1; A29: dom (sn -FanMorphW) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A30: 1 + sn > 0 by A1, XREAL_1:148; 0 <= (p2 `1) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A27, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (p2 `2) / |.p2.| >= - 1 by SQUARE_1:51; then ((p2 `2) / |.p2.|) * (1 + sn) >= (- 1) * (1 + sn) by A30, XREAL_1:64; then ((((p2 `2) / |.p2.|) * (1 + sn)) + sn) - sn >= (- 1) - sn ; then (((p2 `2) / |.p2.|) * (1 + sn)) + sn >= - 1 by XREAL_1:9; then 1 ^2 >= ((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2 by A25, SQUARE_1:49; then A31: 1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2) >= 0 by XREAL_1:48; then A32: sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)) >= 0 by SQUARE_1:def_2; A33: |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `1 = - (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))) by EUCLID:52; then |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 = ((- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))) ^2) + ((|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn)) ^2) by A26, JGRAPH_3:1 .= ((|.p2.| ^2) * ((sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))) ^2)) + ((|.p2.| ^2) * (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)) ; then A34: |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 = ((|.p2.| ^2) * (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))) + ((|.p2.| ^2) * (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)) by A31, SQUARE_1:def_2 .= |.p2.| ^2 ; then A35: |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| = sqrt (|.p2.| ^2) by SQUARE_1:22 .= |.p2.| by SQUARE_1:22 ; then A36: |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| <> 0. (TOP-REAL 2) by A23, TOPRNS_1:23, TOPRNS_1:24; (((p2 `2) / |.p2.|) * (1 + sn)) + sn <= 0 + sn by A23, A24, XREAL_1:7; then (|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| <= sn by A23, A26, A35, TOPRNS_1:24, XCMPLX_1:89; then A37: (sn -FanMorphW) . |[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| = |[(|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (1 - (((((|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) ^2))))),(|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * ((((|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)))]| by A1, A2, A33, A32, A36, Th18; A38: |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (((p2 `1) / |.p2.|) ^2))) = |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (- ((p2 `1) / |.p2.|))) by A23, SQUARE_1:23 .= p2 `1 by A28, A35, XCMPLX_1:87 ; A39: |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * ((((|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) = |.p2.| * ((((((p2 `2) / |.p2.|) * (1 + sn)) + sn) - sn) / (1 + sn)) by A23, A26, A35, TOPRNS_1:24, XCMPLX_1:89 .= |.p2.| * ((p2 `2) / |.p2.|) by A30, XCMPLX_1:89 .= p2 `2 by A23, TOPRNS_1:24, XCMPLX_1:87 ; then |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (1 - (((((|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) ^2)))) = |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (1 - (((p2 `2) / |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) ^2)))) by A23, A35, TOPRNS_1:24, XCMPLX_1:89 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (1 - (((p2 `2) ^2) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2))))) by XCMPLX_1:76 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (((|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2)) - (((p2 `2) ^2) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2))))) by A27, A34, XCMPLX_1:60 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (((|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2) - ((p2 `2) ^2)) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2)))) by XCMPLX_1:120 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (((((p2 `1) ^2) + ((p2 `2) ^2)) - ((p2 `2) ^2)) / (|.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2)))) by A34, JGRAPH_3:1 .= |.|[(- (|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (- (sqrt (((p2 `1) / |.p2.|) ^2))) by A35, XCMPLX_1:76 ; hence ex x being set st ( x in dom (sn -FanMorphW) & y = (sn -FanMorphW) . x ) by A37, A39, A38, A29, EUCLID:53; ::_thesis: verum end; end; end; hence y in rng f by A3, FUNCT_1:def_3; ::_thesis: verum end; hence rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) by A3, XBOOLE_0:def_10; ::_thesis: verum end; hence rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) ; ::_thesis: verum end; Lm12: for q4, q, p2 being Point of (TOP-REAL 2) for O, u, uq being Point of (Euclid 2) st u in cl_Ball (O,(|.p2.| + 1)) & q = uq & q4 = u & O = 0. (TOP-REAL 2) & |.q4.| = |.q.| holds q in cl_Ball (O,(|.p2.| + 1)) proof let q4, q, p2 be Point of (TOP-REAL 2); ::_thesis: for O, u, uq being Point of (Euclid 2) st u in cl_Ball (O,(|.p2.| + 1)) & q = uq & q4 = u & O = 0. (TOP-REAL 2) & |.q4.| = |.q.| holds q in cl_Ball (O,(|.p2.| + 1)) let O, u, uq be Point of (Euclid 2); ::_thesis: ( u in cl_Ball (O,(|.p2.| + 1)) & q = uq & q4 = u & O = 0. (TOP-REAL 2) & |.q4.| = |.q.| implies q in cl_Ball (O,(|.p2.| + 1)) ) assume A1: u in cl_Ball (O,(|.p2.| + 1)) ; ::_thesis: ( not q = uq or not q4 = u or not O = 0. (TOP-REAL 2) or not |.q4.| = |.q.| or q in cl_Ball (O,(|.p2.| + 1)) ) assume that A2: q = uq and A3: q4 = u and A4: O = 0. (TOP-REAL 2) ; ::_thesis: ( not |.q4.| = |.q.| or q in cl_Ball (O,(|.p2.| + 1)) ) assume A5: |.q4.| = |.q.| ; ::_thesis: q in cl_Ball (O,(|.p2.| + 1)) now__::_thesis:_q_in_cl_Ball_(O,(|.p2.|_+_1)) assume not q in cl_Ball (O,(|.p2.| + 1)) ; ::_thesis: contradiction then not dist (O,uq) <= |.p2.| + 1 by A2, METRIC_1:12; then |.((0. (TOP-REAL 2)) - q).| > |.p2.| + 1 by A2, A4, JGRAPH_1:28; then |.(- q).| > |.p2.| + 1 by EUCLID:27; then |.q.| > |.p2.| + 1 by TOPRNS_1:26; then |.(- q4).| > |.p2.| + 1 by A5, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - q4).| > |.p2.| + 1 by EUCLID:27; then dist (O,u) > |.p2.| + 1 by A3, A4, JGRAPH_1:28; hence contradiction by A1, METRIC_1:12; ::_thesis: verum end; hence q in cl_Ball (O,(|.p2.| + 1)) ; ::_thesis: verum end; theorem Th40: :: JGRAPH_4:40 for sn being Real for p2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 holds ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphW) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphW) . p2 in V2 ) ) proof reconsider O = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:67; let sn be Real; ::_thesis: for p2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 holds ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphW) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphW) . p2 in V2 ) ) let p2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 implies ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphW) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphW) . p2 in V2 ) ) ) A1: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider V0 = Ball (O,(|.p2.| + 1)) as Subset of (TOP-REAL 2) ; ( O in V0 & V0 c= cl_Ball (O,(|.p2.| + 1)) ) by GOBOARD6:1, METRIC_1:14; then reconsider K0 = cl_Ball (O,(|.p2.| + 1)) as non empty compact Subset of (TOP-REAL 2) by A1, Th15; set q3 = (sn -FanMorphW) . p2; reconsider VV0 = V0 as Subset of (TopSpaceMetr (Euclid 2)) ; reconsider u2 = p2 as Point of (Euclid 2) by EUCLID:67; reconsider u3 = (sn -FanMorphW) . p2 as Point of (Euclid 2) by EUCLID:67; A2: (sn -FanMorphW) .: K0 c= K0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (sn -FanMorphW) .: K0 or y in K0 ) assume y in (sn -FanMorphW) .: K0 ; ::_thesis: y in K0 then consider x being set such that A3: x in dom (sn -FanMorphW) and A4: x in K0 and A5: y = (sn -FanMorphW) . x by FUNCT_1:def_6; reconsider q = x as Point of (TOP-REAL 2) by A3; reconsider uq = q as Point of (Euclid 2) by EUCLID:67; dist (O,uq) <= |.p2.| + 1 by A4, METRIC_1:12; then |.((0. (TOP-REAL 2)) - q).| <= |.p2.| + 1 by JGRAPH_1:28; then |.(- q).| <= |.p2.| + 1 by EUCLID:27; then A6: |.q.| <= |.p2.| + 1 by TOPRNS_1:26; A7: y in rng (sn -FanMorphW) by A3, A5, FUNCT_1:def_3; then reconsider u = y as Point of (Euclid 2) by EUCLID:67; reconsider q4 = y as Point of (TOP-REAL 2) by A7; |.q4.| = |.q.| by A5, Th33; then |.(- q4).| <= |.p2.| + 1 by A6, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - q4).| <= |.p2.| + 1 by EUCLID:27; then dist (O,u) <= |.p2.| + 1 by JGRAPH_1:28; hence y in K0 by METRIC_1:12; ::_thesis: verum end; VV0 is open by TOPMETR:14; then A8: V0 is open by Lm11, PRE_TOPC:30; A9: |.p2.| < |.p2.| + 1 by XREAL_1:29; then |.(- p2).| < |.p2.| + 1 by TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - p2).| < |.p2.| + 1 by EUCLID:27; then dist (O,u2) < |.p2.| + 1 by JGRAPH_1:28; then A10: p2 in V0 by METRIC_1:11; |.((sn -FanMorphW) . p2).| = |.p2.| by Th33; then |.(- ((sn -FanMorphW) . p2)).| < |.p2.| + 1 by A9, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - ((sn -FanMorphW) . p2)).| < |.p2.| + 1 by EUCLID:27; then dist (O,u3) < |.p2.| + 1 by JGRAPH_1:28; then A11: (sn -FanMorphW) . p2 in V0 by METRIC_1:11; assume A12: ( - 1 < sn & sn < 1 ) ; ::_thesis: ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphW) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphW) . p2 in V2 ) ) K0 c= (sn -FanMorphW) .: K0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in K0 or y in (sn -FanMorphW) .: K0 ) assume A13: y in K0 ; ::_thesis: y in (sn -FanMorphW) .: K0 then reconsider y = y as Point of (Euclid 2) ; reconsider q4 = y as Point of (TOP-REAL 2) by A13; the carrier of (TOP-REAL 2) c= rng (sn -FanMorphW) by A12, Th39; then q4 in rng (sn -FanMorphW) by TARSKI:def_3; then consider x being set such that A14: x in dom (sn -FanMorphW) and A15: y = (sn -FanMorphW) . x by FUNCT_1:def_3; reconsider x = x as Point of (Euclid 2) by A14, Lm11; reconsider q = x as Point of (TOP-REAL 2) by A14; |.q4.| = |.q.| by A15, Th33; then q in K0 by A13, Lm12; hence y in (sn -FanMorphW) .: K0 by A14, A15, FUNCT_1:def_6; ::_thesis: verum end; then K0 = (sn -FanMorphW) .: K0 by A2, XBOOLE_0:def_10; hence ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphW) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphW) . p2 in V2 ) ) by A10, A8, A11, METRIC_1:14; ::_thesis: verum end; theorem :: JGRAPH_4:41 for sn being Real st - 1 < sn & sn < 1 holds ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = sn -FanMorphW & f is being_homeomorphism ) proof let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = sn -FanMorphW & f is being_homeomorphism ) ) reconsider f = sn -FanMorphW as Function of (TOP-REAL 2),(TOP-REAL 2) ; assume A1: ( - 1 < sn & sn < 1 ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = sn -FanMorphW & f is being_homeomorphism ) then A2: for p2 being Point of (TOP-REAL 2) ex K being non empty compact Subset of (TOP-REAL 2) st ( K = f .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & f . p2 in V2 ) ) by Th40; ( rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) & ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphW & h is continuous ) ) by A1, Th37, Th39; then f is being_homeomorphism by A1, A2, Th3, Th38; hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = sn -FanMorphW & f is being_homeomorphism ) ; ::_thesis: verum end; Lm13: now__::_thesis:_for_q_being_Point_of_(TOP-REAL_2) for_sn,_t_being_Real_st_((-_((t_/_|.q.|)_-_sn))_/_(1_-_sn))_^2_<_1_^2_holds_ -_(sqrt_(1_-_((((t_/_|.q.|)_-_sn)_/_(1_-_sn))_^2)))_<_-_0 let q be Point of (TOP-REAL 2); ::_thesis: for sn, t being Real st ((- ((t / |.q.|) - sn)) / (1 - sn)) ^2 < 1 ^2 holds - (sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2))) < - 0 let sn, t be Real; ::_thesis: ( ((- ((t / |.q.|) - sn)) / (1 - sn)) ^2 < 1 ^2 implies - (sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2))) < - 0 ) assume ((- ((t / |.q.|) - sn)) / (1 - sn)) ^2 < 1 ^2 ; ::_thesis: - (sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2))) < - 0 then 1 - (((- ((t / |.q.|) - sn)) / (1 - sn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((- ((t / |.q.|) - sn)) / (1 - sn)) ^2)) > 0 by SQUARE_1:25; then sqrt (1 - (((- ((t / |.q.|) - sn)) ^2) / ((1 - sn) ^2))) > 0 by XCMPLX_1:76; then sqrt (1 - ((((t / |.q.|) - sn) ^2) / ((1 - sn) ^2))) > 0 ; then sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2)) > 0 by XCMPLX_1:76; hence - (sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2))) < - 0 by XREAL_1:24; ::_thesis: verum end; theorem Th42: :: JGRAPH_4:42 for sn being Real for q being Point of (TOP-REAL 2) st sn < 1 & q `1 < 0 & (q `2) / |.q.| >= sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 >= 0 ) proof let sn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st sn < 1 & q `1 < 0 & (q `2) / |.q.| >= sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 >= 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( sn < 1 & q `1 < 0 & (q `2) / |.q.| >= sn implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 >= 0 ) ) assume that A1: sn < 1 and A2: q `1 < 0 and A3: (q `2) / |.q.| >= sn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 >= 0 ) A4: 1 - sn > 0 by A1, XREAL_1:149; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (sn -FanMorphW) . q implies ( p `1 < 0 & p `2 >= 0 ) ) set qz = p; assume p = (sn -FanMorphW) . q ; ::_thesis: ( p `1 < 0 & p `2 >= 0 ) then A5: p = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by A2, A3, Th16; then A6: p `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; A7: ((q `2) / |.q.|) - sn >= 0 by A3, XREAL_1:48; A8: |.q.| > 0 by A2, Lm1, JGRAPH_2:3; then A9: |.q.| ^2 > 0 by SQUARE_1:12; ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) < ((q `1) ^2) + ((q `2) ^2) ) by A2, JGRAPH_3:1, SQUARE_1:12, XREAL_1:8; then ((q `2) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A9, XREAL_1:74; then ((q `2) ^2) / (|.q.| ^2) < 1 by A9, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 < 1 by XCMPLX_1:76; then 1 > (q `2) / |.q.| by SQUARE_1:52; then 1 - sn > ((q `2) / |.q.|) - sn by XREAL_1:9; then - (1 - sn) < - (((q `2) / |.q.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) < (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A4, XREAL_1:74; then - 1 < (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A4, XCMPLX_1:197; then ((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2 < 1 ^2 by A4, A7, SQUARE_1:50; hence ( p `1 < 0 & p `2 >= 0 ) by A5, A8, A4, A6, A7, Lm13, EUCLID:52, XREAL_1:132; ::_thesis: verum end; theorem Th43: :: JGRAPH_4:43 for sn being Real for q being Point of (TOP-REAL 2) st - 1 < sn & q `1 < 0 & (q `2) / |.q.| < sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 < 0 ) proof let sn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < sn & q `1 < 0 & (q `2) / |.q.| < sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 < 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & q `1 < 0 & (q `2) / |.q.| < sn implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 < 0 ) ) assume that A1: - 1 < sn and A2: q `1 < 0 and A3: (q `2) / |.q.| < sn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 < 0 ) A4: 1 + sn > 0 by A1, XREAL_1:148; A5: ((q `2) / |.q.|) - sn < 0 by A3, XREAL_1:49; then - (((q `2) / |.q.|) - sn) > 0 by XREAL_1:58; then (- (1 + sn)) / (1 + sn) < (- (((q `2) / |.q.|) - sn)) / (1 + sn) by A4, XREAL_1:74; then A6: - 1 < (- (((q `2) / |.q.|) - sn)) / (1 + sn) by A4, XCMPLX_1:197; |.q.| > 0 by A2, Lm1, JGRAPH_2:3; then A7: |.q.| ^2 > 0 by SQUARE_1:12; ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) < ((q `1) ^2) + ((q `2) ^2) ) by A2, JGRAPH_3:1, SQUARE_1:12, XREAL_1:8; then ((q `2) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A7, XREAL_1:74; then ((q `2) ^2) / (|.q.| ^2) < 1 by A7, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (q `2) / |.q.| by SQUARE_1:52; then (- 1) - sn < ((q `2) / |.q.|) - sn by XREAL_1:9; then - (- (1 + sn)) > - (((q `2) / |.q.|) - sn) by XREAL_1:24; then (- (((q `2) / |.q.|) - sn)) / (1 + sn) < 1 by A4, XREAL_1:191; then ((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2 < 1 ^2 by A6, SQUARE_1:50; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2)) > 0 by SQUARE_1:25; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) ^2) / ((1 + sn) ^2))) > 0 by XCMPLX_1:76; then sqrt (1 - (((((q `2) / |.q.|) - sn) ^2) / ((1 + sn) ^2))) > 0 ; then sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) > 0 by XCMPLX_1:76; then A8: - (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) < - 0 by XREAL_1:24; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (sn -FanMorphW) . q implies ( p `1 < 0 & p `2 < 0 ) ) set qz = p; assume p = (sn -FanMorphW) . q ; ::_thesis: ( p `1 < 0 & p `2 < 0 ) then p = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A2, A3, Th17; then A9: ( p `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))) & p `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) ) by EUCLID:52; (((q `2) / |.q.|) - sn) / (1 + sn) < 0 by A1, A5, XREAL_1:141, XREAL_1:148; hence ( p `1 < 0 & p `2 < 0 ) by A2, A9, A8, Lm1, JGRAPH_2:3, XREAL_1:132; ::_thesis: verum end; theorem Th44: :: JGRAPH_4:44 for sn being Real for q1, q2 being Point of (TOP-REAL 2) st sn < 1 & q1 `1 < 0 & (q1 `2) / |.q1.| >= sn & q2 `1 < 0 & (q2 `2) / |.q2.| >= sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| proof let sn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st sn < 1 & q1 `1 < 0 & (q1 `2) / |.q1.| >= sn & q2 `1 < 0 & (q2 `2) / |.q2.| >= sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( sn < 1 & q1 `1 < 0 & (q1 `2) / |.q1.| >= sn & q2 `1 < 0 & (q2 `2) / |.q2.| >= sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A1: sn < 1 and A2: q1 `1 < 0 and A3: (q1 `2) / |.q1.| >= sn and A4: q2 `1 < 0 and A5: (q2 `2) / |.q2.| >= sn and A6: (q1 `2) / |.q1.| < (q2 `2) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| A7: ( ((q1 `2) / |.q1.|) - sn < ((q2 `2) / |.q2.|) - sn & 1 - sn > 0 ) by A1, A6, XREAL_1:9, XREAL_1:149; let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 implies (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A8: p1 = (sn -FanMorphW) . q1 and A9: p2 = (sn -FanMorphW) . q2 ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| A10: |.p2.| = |.q2.| by A9, Th33; p2 = |[(|.q2.| * (- (sqrt (1 - (((((q2 `2) / |.q2.|) - sn) / (1 - sn)) ^2))))),(|.q2.| * ((((q2 `2) / |.q2.|) - sn) / (1 - sn)))]| by A4, A5, A9, Th16; then A11: p2 `2 = |.q2.| * ((((q2 `2) / |.q2.|) - sn) / (1 - sn)) by EUCLID:52; |.q2.| > 0 by A4, Lm1, JGRAPH_2:3; then A12: (p2 `2) / |.p2.| = (((q2 `2) / |.q2.|) - sn) / (1 - sn) by A11, A10, XCMPLX_1:89; p1 = |[(|.q1.| * (- (sqrt (1 - (((((q1 `2) / |.q1.|) - sn) / (1 - sn)) ^2))))),(|.q1.| * ((((q1 `2) / |.q1.|) - sn) / (1 - sn)))]| by A2, A3, A8, Th16; then A13: p1 `2 = |.q1.| * ((((q1 `2) / |.q1.|) - sn) / (1 - sn)) by EUCLID:52; A14: |.p1.| = |.q1.| by A8, Th33; |.q1.| > 0 by A2, Lm1, JGRAPH_2:3; then (p1 `2) / |.p1.| = (((q1 `2) / |.q1.|) - sn) / (1 - sn) by A13, A14, XCMPLX_1:89; hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A12, A7, XREAL_1:74; ::_thesis: verum end; theorem Th45: :: JGRAPH_4:45 for sn being Real for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & q1 `1 < 0 & (q1 `2) / |.q1.| < sn & q2 `1 < 0 & (q2 `2) / |.q2.| < sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| proof let sn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & q1 `1 < 0 & (q1 `2) / |.q1.| < sn & q2 `1 < 0 & (q2 `2) / |.q2.| < sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & q1 `1 < 0 & (q1 `2) / |.q1.| < sn & q2 `1 < 0 & (q2 `2) / |.q2.| < sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A1: - 1 < sn and A2: q1 `1 < 0 and A3: (q1 `2) / |.q1.| < sn and A4: q2 `1 < 0 and A5: (q2 `2) / |.q2.| < sn and A6: (q1 `2) / |.q1.| < (q2 `2) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| A7: ( ((q1 `2) / |.q1.|) - sn < ((q2 `2) / |.q2.|) - sn & 1 + sn > 0 ) by A1, A6, XREAL_1:9, XREAL_1:148; let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 implies (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A8: p1 = (sn -FanMorphW) . q1 and A9: p2 = (sn -FanMorphW) . q2 ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| A10: |.p2.| = |.q2.| by A9, Th33; p2 = |[(|.q2.| * (- (sqrt (1 - (((((q2 `2) / |.q2.|) - sn) / (1 + sn)) ^2))))),(|.q2.| * ((((q2 `2) / |.q2.|) - sn) / (1 + sn)))]| by A4, A5, A9, Th17; then A11: p2 `2 = |.q2.| * ((((q2 `2) / |.q2.|) - sn) / (1 + sn)) by EUCLID:52; |.q2.| > 0 by A4, Lm1, JGRAPH_2:3; then A12: (p2 `2) / |.p2.| = (((q2 `2) / |.q2.|) - sn) / (1 + sn) by A11, A10, XCMPLX_1:89; p1 = |[(|.q1.| * (- (sqrt (1 - (((((q1 `2) / |.q1.|) - sn) / (1 + sn)) ^2))))),(|.q1.| * ((((q1 `2) / |.q1.|) - sn) / (1 + sn)))]| by A2, A3, A8, Th17; then A13: p1 `2 = |.q1.| * ((((q1 `2) / |.q1.|) - sn) / (1 + sn)) by EUCLID:52; A14: |.p1.| = |.q1.| by A8, Th33; |.q1.| > 0 by A2, Lm1, JGRAPH_2:3; then (p1 `2) / |.p1.| = (((q1 `2) / |.q1.|) - sn) / (1 + sn) by A13, A14, XCMPLX_1:89; hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A12, A7, XREAL_1:74; ::_thesis: verum end; theorem :: JGRAPH_4:46 for sn being Real for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q1 `1 < 0 & q2 `1 < 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| proof let sn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q1 `1 < 0 & q2 `1 < 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 & q1 `1 < 0 & q2 `1 < 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A1: - 1 < sn and A2: sn < 1 and A3: q1 `1 < 0 and A4: q2 `1 < 0 and A5: (q1 `2) / |.q1.| < (q2 `2) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 implies (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A6: p1 = (sn -FanMorphW) . q1 and A7: p2 = (sn -FanMorphW) . q2 ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| now__::_thesis:_(_(_(q1_`2)_/_|.q1.|_>=_sn_&_(q2_`2)_/_|.q2.|_>=_sn_&_(p1_`2)_/_|.p1.|_<_(p2_`2)_/_|.p2.|_)_or_(_(q1_`2)_/_|.q1.|_>=_sn_&_(q2_`2)_/_|.q2.|_<_sn_&_(p1_`2)_/_|.p1.|_<_(p2_`2)_/_|.p2.|_)_or_(_(q1_`2)_/_|.q1.|_<_sn_&_(q2_`2)_/_|.q2.|_>=_sn_&_(p1_`2)_/_|.p1.|_<_(p2_`2)_/_|.p2.|_)_or_(_(q1_`2)_/_|.q1.|_<_sn_&_(q2_`2)_/_|.q2.|_<_sn_&_(p1_`2)_/_|.p1.|_<_(p2_`2)_/_|.p2.|_)_) percases ( ( (q1 `2) / |.q1.| >= sn & (q2 `2) / |.q2.| >= sn ) or ( (q1 `2) / |.q1.| >= sn & (q2 `2) / |.q2.| < sn ) or ( (q1 `2) / |.q1.| < sn & (q2 `2) / |.q2.| >= sn ) or ( (q1 `2) / |.q1.| < sn & (q2 `2) / |.q2.| < sn ) ) ; case ( (q1 `2) / |.q1.| >= sn & (q2 `2) / |.q2.| >= sn ) ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A2, A3, A4, A5, A6, A7, Th44; ::_thesis: verum end; case ( (q1 `2) / |.q1.| >= sn & (q2 `2) / |.q2.| < sn ) ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A5, XXREAL_0:2; ::_thesis: verum end; caseA8: ( (q1 `2) / |.q1.| < sn & (q2 `2) / |.q2.| >= sn ) ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| then p2 `2 >= 0 by A2, A4, A7, Th42; then A9: (p2 `2) / |.p2.| >= 0 ; p1 `2 < 0 by A1, A3, A6, A8, Th43; hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A9, Lm1, JGRAPH_2:3, XREAL_1:141; ::_thesis: verum end; case ( (q1 `2) / |.q1.| < sn & (q2 `2) / |.q2.| < sn ) ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A1, A3, A4, A5, A6, A7, Th45; ::_thesis: verum end; end; end; hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ; ::_thesis: verum end; theorem :: JGRAPH_4:47 for sn being Real for q being Point of (TOP-REAL 2) st q `1 < 0 & (q `2) / |.q.| = sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 = 0 ) proof let sn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st q `1 < 0 & (q `2) / |.q.| = sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 = 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( q `1 < 0 & (q `2) / |.q.| = sn implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 = 0 ) ) assume that A1: q `1 < 0 and A2: (q `2) / |.q.| = sn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds ( p `1 < 0 & p `2 = 0 ) let p be Point of (TOP-REAL 2); ::_thesis: ( p = (sn -FanMorphW) . q implies ( p `1 < 0 & p `2 = 0 ) ) A3: |.q.| > 0 by A1, Lm1, JGRAPH_2:3; assume p = (sn -FanMorphW) . q ; ::_thesis: ( p `1 < 0 & p `2 = 0 ) then A4: p = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, Th16; then p `1 = |.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))) by EUCLID:52; hence ( p `1 < 0 & p `2 = 0 ) by A2, A4, A3, Lm13, EUCLID:52, XREAL_1:132; ::_thesis: verum end; theorem :: JGRAPH_4:48 for sn being real number holds 0. (TOP-REAL 2) = (sn -FanMorphW) . (0. (TOP-REAL 2)) by Th16, JGRAPH_2:3; begin definition let s be real number ; let q be Point of (TOP-REAL 2); func FanN (s,q) -> Point of (TOP-REAL 2) equals :Def4: :: JGRAPH_4:def 4 |.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2)))]| if ( (q `1) / |.q.| >= s & q `2 > 0 ) |.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2)))]| if ( (q `1) / |.q.| < s & q `2 > 0 ) otherwise q; correctness coherence ( ( (q `1) / |.q.| >= s & q `2 > 0 implies |.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2)))]| is Point of (TOP-REAL 2) ) & ( (q `1) / |.q.| < s & q `2 > 0 implies |.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2)))]| is Point of (TOP-REAL 2) ) & ( ( not (q `1) / |.q.| >= s or not q `2 > 0 ) & ( not (q `1) / |.q.| < s or not q `2 > 0 ) implies q is Point of (TOP-REAL 2) ) ); consistency for b1 being Point of (TOP-REAL 2) st (q `1) / |.q.| >= s & q `2 > 0 & (q `1) / |.q.| < s & q `2 > 0 holds ( b1 = |.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2)))]| iff b1 = |.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2)))]| ); ; end; :: deftheorem Def4 defines FanN JGRAPH_4:def_4_:_ for s being real number for q being Point of (TOP-REAL 2) holds ( ( (q `1) / |.q.| >= s & q `2 > 0 implies FanN (s,q) = |.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2)))]| ) & ( (q `1) / |.q.| < s & q `2 > 0 implies FanN (s,q) = |.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2)))]| ) & ( ( not (q `1) / |.q.| >= s or not q `2 > 0 ) & ( not (q `1) / |.q.| < s or not q `2 > 0 ) implies FanN (s,q) = q ) ); definition let c be real number ; funcc -FanMorphN -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def5: :: JGRAPH_4:def 5 for q being Point of (TOP-REAL 2) holds it . q = FanN (c,q); existence ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st for q being Point of (TOP-REAL 2) holds b1 . q = FanN (c,q) proof deffunc H1( Point of (TOP-REAL 2)) -> Point of (TOP-REAL 2) = FanN (c,$1); thus ex IT being Function of (TOP-REAL 2),(TOP-REAL 2) st for q being Point of (TOP-REAL 2) holds IT . q = H1(q) from FUNCT_2:sch_4(); ::_thesis: verum end; uniqueness for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for q being Point of (TOP-REAL 2) holds b1 . q = FanN (c,q) ) & ( for q being Point of (TOP-REAL 2) holds b2 . q = FanN (c,q) ) holds b1 = b2 proof deffunc H1( Point of (TOP-REAL 2)) -> Point of (TOP-REAL 2) = FanN (c,$1); thus for a, b being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for q being Point of (TOP-REAL 2) holds a . q = H1(q) ) & ( for q being Point of (TOP-REAL 2) holds b . q = H1(q) ) holds a = b from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def5 defines -FanMorphN JGRAPH_4:def_5_:_ for c being real number for b2 being Function of (TOP-REAL 2),(TOP-REAL 2) holds ( b2 = c -FanMorphN iff for q being Point of (TOP-REAL 2) holds b2 . q = FanN (c,q) ); theorem Th49: :: JGRAPH_4:49 for q being Point of (TOP-REAL 2) for cn being real number holds ( ( (q `1) / |.q.| >= cn & q `2 > 0 implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( q `2 <= 0 implies (cn -FanMorphN) . q = q ) ) proof let q be Point of (TOP-REAL 2); ::_thesis: for cn being real number holds ( ( (q `1) / |.q.| >= cn & q `2 > 0 implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( q `2 <= 0 implies (cn -FanMorphN) . q = q ) ) let cn be real number ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 > 0 implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( q `2 <= 0 implies (cn -FanMorphN) . q = q ) ) hereby ::_thesis: ( q `2 <= 0 implies (cn -FanMorphN) . q = q ) assume ( (q `1) / |.q.| >= cn & q `2 > 0 ) ; ::_thesis: (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| then FanN (cn,q) = |.q.| * |[((((q `1) / |.q.|) - cn) / (1 - cn)),(sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))]| by Def4 .= |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by EUCLID:58 ; hence (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by Def5; ::_thesis: verum end; assume A1: q `2 <= 0 ; ::_thesis: (cn -FanMorphN) . q = q (cn -FanMorphN) . q = FanN (cn,q) by Def5; hence (cn -FanMorphN) . q = q by A1, Def4; ::_thesis: verum end; theorem Th50: :: JGRAPH_4:50 for q being Point of (TOP-REAL 2) for cn being Real st (q `1) / |.q.| <= cn & q `2 > 0 holds (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| proof let q be Point of (TOP-REAL 2); ::_thesis: for cn being Real st (q `1) / |.q.| <= cn & q `2 > 0 holds (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| let cn be Real; ::_thesis: ( (q `1) / |.q.| <= cn & q `2 > 0 implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) assume that A1: (q `1) / |.q.| <= cn and A2: q `2 > 0 ; ::_thesis: (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| percases ( (q `1) / |.q.| < cn or (q `1) / |.q.| = cn ) by A1, XXREAL_0:1; suppose (q `1) / |.q.| < cn ; ::_thesis: (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| then FanN (cn,q) = |.q.| * |[((((q `1) / |.q.|) - cn) / (1 + cn)),(sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))]| by A2, Def4 .= |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| by EUCLID:58 ; hence (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| by Def5; ::_thesis: verum end; supposeA3: (q `1) / |.q.| = cn ; ::_thesis: (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| then (((q `1) / |.q.|) - cn) / (1 - cn) = 0 ; hence (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| by A2, A3, Th49; ::_thesis: verum end; end; end; theorem Th51: :: JGRAPH_4:51 for q being Point of (TOP-REAL 2) for cn being Real st - 1 < cn & cn < 1 holds ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) proof let q be Point of (TOP-REAL 2); ::_thesis: for cn being Real st - 1 < cn & cn < 1 holds ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) ) assume that A1: - 1 < cn and A2: cn < 1 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) percases ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) ) or ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) ) or q `2 < 0 or q = 0. (TOP-REAL 2) ) ; supposeA3: ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) percases ( q `2 > 0 or q `2 <= 0 ) ; supposeA4: q `2 > 0 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) then FanN (cn,q) = |.q.| * |[((((q `1) / |.q.|) - cn) / (1 - cn)),(sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))]| by A3, Def4 .= |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by EUCLID:58 ; hence ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) by A4, Def5, Th50; ::_thesis: verum end; supposeA5: q `2 <= 0 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) then A6: (cn -FanMorphN) . q = q by Th49; A7: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; A8: 1 - cn > 0 by A2, XREAL_1:149; A9: q `2 = 0 by A3, A5; |.q.| <> 0 by A3, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `1) ^2) / (|.q.| ^2) = 1 ^2 by A7, A9, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 = 1 ^2 by XCMPLX_1:76; then A10: sqrt (((q `1) / |.q.|) ^2) = 1 by SQUARE_1:22; A11: now__::_thesis:_not_q_`1_<_0 assume q `1 < 0 ; ::_thesis: contradiction then - ((q `1) / |.q.|) = 1 by A10, SQUARE_1:23; hence contradiction by A1, A3; ::_thesis: verum end; sqrt (|.q.| ^2) = |.q.| by SQUARE_1:22; then A12: |.q.| = q `1 by A7, A9, A11, SQUARE_1:22; then 1 = (q `1) / |.q.| by A3, TOPRNS_1:24, XCMPLX_1:60; then (((q `1) / |.q.|) - cn) / (1 - cn) = 1 by A8, XCMPLX_1:60; hence ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) by A2, A6, A9, A12, EUCLID:53, SQUARE_1:17, TOPRNS_1:24, XCMPLX_1:60; ::_thesis: verum end; end; end; supposeA13: ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) percases ( q `2 > 0 or q `2 <= 0 ) ; suppose q `2 > 0 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) hence ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) by Th49, Th50; ::_thesis: verum end; supposeA14: q `2 <= 0 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) A15: 1 + cn > 0 by A1, XREAL_1:148; A16: |.q.| <> 0 by A13, TOPRNS_1:24; A17: q `2 = 0 by A13, A14; ( |.q.| > 0 & 1 > (q `1) / |.q.| ) by A2, A13, Lm1, XXREAL_0:2; then 1 * |.q.| > ((q `1) / |.q.|) * |.q.| by XREAL_1:68; then A18: ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & |.q.| > q `1 ) by A13, JGRAPH_3:1, TOPRNS_1:24, XCMPLX_1:87; then A19: q `1 = - |.q.| by A17, SQUARE_1:40; then - 1 = (q `1) / |.q.| by A13, TOPRNS_1:24, XCMPLX_1:197; then A20: (((q `1) / |.q.|) - cn) / (1 + cn) = (- (1 + cn)) / (1 + cn) .= - 1 by A15, XCMPLX_1:197 ; |.q.| = - (q `1) by A17, A18, SQUARE_1:40; then |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| = q by A17, A20, EUCLID:53, SQUARE_1:17; hence ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) by A1, A14, A16, A19, Th49, XCMPLX_1:197; ::_thesis: verum end; end; end; suppose ( q `2 < 0 or q = 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) hence ( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) ) ; ::_thesis: verum end; end; end; theorem Th52: :: JGRAPH_4:52 for cn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let cn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = cn; set b = 1 - cn; reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2; assume that A1: cn < 1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 - cn > 0 by A1, XREAL_1:149; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (((r1 / r2) - cn) / (1 - cn)) and A6: g3 is continuous by A4, Th5; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in K1 by A7, A8, A9, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( proj1 . r = r `1 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_5; A11: ( g2 . s = proj1 . s & g1 . s = (2 NormF) . s ) by Lm2, Lm5; f . r = |.r.| * ((((r `1) / |.r.|) - cn) / (1 - cn)) by A2, A9; hence f . x = g3 . x by A5, A11, A10; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th53: :: JGRAPH_4:53 for cn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let cn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = cn; set b = 1 + cn; reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2; assume that A1: - 1 < cn and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 + cn > 0 by A1, XREAL_1:148; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (((r1 / r2) - cn) / (1 + cn)) and A6: g3 is continuous by A4, Th5; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; A8: for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in dom g3 by A7, A9; then x in K1 by A7, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( proj1 . r = r `1 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_5; A11: ( g2 . s = proj1 . s & g1 . s = (2 NormF) . s ) by Lm2, Lm5; f . r = |.r.| * ((((r `1) / |.r.|) - cn) / (1 + cn)) by A2, A9; hence f . x = g3 . x by A5, A11, A10; ::_thesis: verum end; dom f = dom g3 by A7, FUNCT_2:def_1; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th54: :: JGRAPH_4:54 for cn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let cn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = cn; set b = 1 - cn; reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2; assume that A1: cn < 1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 - cn > 0 by A1, XREAL_1:149; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (sqrt (abs (1 - ((((r1 / r2) - cn) / (1 - cn)) ^2)))) and A6: g3 is continuous by A4, Th10; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) A9: 1 - cn > 0 by A1, XREAL_1:149; assume A10: x in dom f ; ::_thesis: f . x = g3 . x then x in K1 by A7, A8, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A11: |.r.| <> 0 by A3, A10, TOPRNS_1:24; |.r.| ^2 = ((r `1) ^2) + ((r `2) ^2) by JGRAPH_3:1; then A12: ((r `1) - |.r.|) * ((r `1) + |.r.|) = - ((r `2) ^2) ; (r `2) ^2 >= 0 by XREAL_1:63; then r `1 <= |.r.| by A12, XREAL_1:93; then (r `1) / |.r.| <= |.r.| / |.r.| by XREAL_1:72; then (r `1) / |.r.| <= 1 by A11, XCMPLX_1:60; then A13: ((r `1) / |.r.|) - cn <= 1 - cn by XREAL_1:9; reconsider s = x as Point of ((TOP-REAL 2) | K1) by A10; A14: now__::_thesis:_not_(1_-_cn)_^2_=_0 assume (1 - cn) ^2 = 0 ; ::_thesis: contradiction then (1 - cn) + cn = 0 + cn by XCMPLX_1:6; hence contradiction by A1; ::_thesis: verum end; cn - ((r `1) / |.r.|) <= 0 by A3, A10, XREAL_1:47; then - (cn - ((r `1) / |.r.|)) >= - (1 - cn) by A9, XREAL_1:24; then ( (1 - cn) ^2 >= 0 & (((r `1) / |.r.|) - cn) ^2 <= (1 - cn) ^2 ) by A13, SQUARE_1:49, XREAL_1:63; then ((((r `1) / |.r.|) - cn) ^2) / ((1 - cn) ^2) <= ((1 - cn) ^2) / ((1 - cn) ^2) by XREAL_1:72; then ((((r `1) / |.r.|) - cn) ^2) / ((1 - cn) ^2) <= 1 by A14, XCMPLX_1:60; then ((((r `1) / |.r.|) - cn) / (1 - cn)) ^2 <= 1 by XCMPLX_1:76; then 1 - (((((r `1) / |.r.|) - cn) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then abs (1 - (((((r `1) / |.r.|) - cn) / (1 - cn)) ^2)) = 1 - (((((r `1) / |.r.|) - cn) / (1 - cn)) ^2) by ABSVALUE:def_1; then A15: f . r = |.r.| * (sqrt (abs (1 - (((((r `1) / |.r.|) - cn) / (1 - cn)) ^2)))) by A2, A10; A16: ( proj1 . r = r `1 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_5; ( g2 . s = proj1 . s & g1 . s = (2 NormF) . s ) by Lm2, Lm5; hence f . x = g3 . x by A5, A15, A16; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th55: :: JGRAPH_4:55 for cn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let cn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = cn; set b = 1 + cn; reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2; assume that A1: - 1 < cn and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous A4: 1 + cn > 0 by A1, XREAL_1:148; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (sqrt (abs (1 - ((((r1 / r2) - cn) / (1 + cn)) ^2)))) and A6: g3 is continuous by A4, Th10; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then x in K1 by A7, A8, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; reconsider s = x as Point of ((TOP-REAL 2) | K1) by A9; A10: (1 + cn) ^2 > 0 by A4, SQUARE_1:12; A11: |.r.| <> 0 by A3, A9, TOPRNS_1:24; |.r.| ^2 = ((r `1) ^2) + ((r `2) ^2) by JGRAPH_3:1; then A12: ((r `1) - |.r.|) * ((r `1) + |.r.|) = - ((r `2) ^2) ; (r `2) ^2 >= 0 by XREAL_1:63; then - |.r.| <= r `1 by A12, XREAL_1:93; then (r `1) / |.r.| >= (- |.r.|) / |.r.| by XREAL_1:72; then (r `1) / |.r.| >= - 1 by A11, XCMPLX_1:197; then ((r `1) / |.r.|) - cn >= (- 1) - cn by XREAL_1:9; then A13: ((r `1) / |.r.|) - cn >= - (1 + cn) ; cn - ((r `1) / |.r.|) >= 0 by A3, A9, XREAL_1:48; then - (cn - ((r `1) / |.r.|)) <= - 0 ; then (((r `1) / |.r.|) - cn) ^2 <= (1 + cn) ^2 by A4, A13, SQUARE_1:49; then ((((r `1) / |.r.|) - cn) ^2) / ((1 + cn) ^2) <= ((1 + cn) ^2) / ((1 + cn) ^2) by A4, XREAL_1:72; then ((((r `1) / |.r.|) - cn) ^2) / ((1 + cn) ^2) <= 1 by A10, XCMPLX_1:60; then ((((r `1) / |.r.|) - cn) / (1 + cn)) ^2 <= 1 by XCMPLX_1:76; then 1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then abs (1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2)) = 1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2) by ABSVALUE:def_1; then A14: f . r = |.r.| * (sqrt (abs (1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2)))) by A2, A9; A15: ( proj1 . r = r `1 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_5; ( g2 . s = proj1 . s & g1 . s = (2 NormF) . s ) by Lm2, Lm5; hence f . x = g3 . x by A5, A14, A15; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th56: :: JGRAPH_4:56 for cn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set sn = sqrt (1 - (cn ^2)); set p0 = |[cn,(sqrt (1 - (cn ^2)))]|; A1: |[cn,(sqrt (1 - (cn ^2)))]| `2 = sqrt (1 - (cn ^2)) by EUCLID:52; |[cn,(sqrt (1 - (cn ^2)))]| `1 = cn by EUCLID:52; then A2: |.|[cn,(sqrt (1 - (cn ^2)))]|.| = sqrt (((sqrt (1 - (cn ^2))) ^2) + (cn ^2)) by A1, JGRAPH_3:1; assume A3: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then cn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (cn ^2) > 0 by XREAL_1:50; then (sqrt (1 - (cn ^2))) ^2 = 1 - (cn ^2) by SQUARE_1:def_2; then A5: (|[cn,(sqrt (1 - (cn ^2)))]| `1) / |.|[cn,(sqrt (1 - (cn ^2)))]|.| = cn by A2, EUCLID:52, SQUARE_1:18; |[cn,(sqrt (1 - (cn ^2)))]| `2 > 0 by A1, A4, SQUARE_1:25; then A6: |[cn,(sqrt (1 - (cn ^2)))]| in K0 by A3, A5, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A7: rng (proj2 * ((cn -FanMorphN) | K1)) c= the carrier of R^1 by TOPMETR:17; A8: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & (p8 `1) / |.p8.| >= cn & p8 `2 >= 0 & p8 <> 0. (TOP-REAL 2) ) by A3; hence x in B0 by A3; ::_thesis: verum end; A9: dom ((cn -FanMorphN) | K1) c= dom (proj1 * ((cn -FanMorphN) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((cn -FanMorphN) | K1) or x in dom (proj1 * ((cn -FanMorphN) | K1)) ) assume A10: x in dom ((cn -FanMorphN) | K1) ; ::_thesis: x in dom (proj1 * ((cn -FanMorphN) | K1)) then x in (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61; then x in dom (cn -FanMorphN) by XBOOLE_0:def_4; then A11: ( dom proj1 = the carrier of (TOP-REAL 2) & (cn -FanMorphN) . x in rng (cn -FanMorphN) ) by FUNCT_1:3, FUNCT_2:def_1; ((cn -FanMorphN) | K1) . x = (cn -FanMorphN) . x by A10, FUNCT_1:47; hence x in dom (proj1 * ((cn -FanMorphN) | K1)) by A10, A11, FUNCT_1:11; ::_thesis: verum end; A12: rng (proj1 * ((cn -FanMorphN) | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1 * ((cn -FanMorphN) | K1)) c= dom ((cn -FanMorphN) | K1) by RELAT_1:25; then dom (proj1 * ((cn -FanMorphN) | K1)) = dom ((cn -FanMorphN) | K1) by A9, XBOOLE_0:def_10 .= (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g2 = proj1 * ((cn -FanMorphN) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A12, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) A13: dom ((cn -FanMorphN) | K1) = (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A14: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A15: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A14; then A16: (cn -FanMorphN) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))))]| by A3, Th51; ((cn -FanMorphN) | K1) . p = (cn -FanMorphN) . p by A15, A14, FUNCT_1:49; then g2 . p = proj1 . |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))))]| by A15, A13, A14, A16, FUNCT_1:13 .= |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))))]| `1 by PSCOMP_1:def_5 .= |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) by EUCLID:52 ; hence g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A17: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ; A18: dom ((cn -FanMorphN) | K1) c= dom (proj2 * ((cn -FanMorphN) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((cn -FanMorphN) | K1) or x in dom (proj2 * ((cn -FanMorphN) | K1)) ) assume A19: x in dom ((cn -FanMorphN) | K1) ; ::_thesis: x in dom (proj2 * ((cn -FanMorphN) | K1)) then x in (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61; then x in dom (cn -FanMorphN) by XBOOLE_0:def_4; then A20: ( dom proj2 = the carrier of (TOP-REAL 2) & (cn -FanMorphN) . x in rng (cn -FanMorphN) ) by FUNCT_1:3, FUNCT_2:def_1; ((cn -FanMorphN) | K1) . x = (cn -FanMorphN) . x by A19, FUNCT_1:47; hence x in dom (proj2 * ((cn -FanMorphN) | K1)) by A19, A20, FUNCT_1:11; ::_thesis: verum end; dom (proj2 * ((cn -FanMorphN) | K1)) c= dom ((cn -FanMorphN) | K1) by RELAT_1:25; then dom (proj2 * ((cn -FanMorphN) | K1)) = dom ((cn -FanMorphN) | K1) by A18, XBOOLE_0:def_10 .= (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g1 = proj2 * ((cn -FanMorphN) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A7, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ) A21: dom ((cn -FanMorphN) | K1) = (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A22: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A23: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A22; then A24: (cn -FanMorphN) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))))]| by A3, Th51; ((cn -FanMorphN) | K1) . p = (cn -FanMorphN) . p by A23, A22, FUNCT_1:49; then g1 . p = proj2 . |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))))]| by A23, A21, A22, A24, FUNCT_1:13 .= |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))))]| `2 by PSCOMP_1:def_6 .= |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) by EUCLID:52 ; hence g1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A25: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) A26: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A26; hence ( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then A27: f1 is continuous by A3, A25, Th54; A28: for x, y, s, r being real number st |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| holds f . |[x,y]| = |[s,r]| proof let x, y, s, r be real number ; ::_thesis: ( |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| implies f . |[x,y]| = |[s,r]| ) assume that A29: |[x,y]| in K1 and A30: ( s = f2 . |[x,y]| & r = f1 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[s,r]| set p99 = |[x,y]|; A31: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A29; A32: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then A33: f1 . |[x,y]| = |.|[x,y]|.| * (sqrt (1 - (((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 - cn)) ^2))) by A25, A29; ((cn -FanMorphN) | K0) . |[x,y]| = (cn -FanMorphN) . |[x,y]| by A29, FUNCT_1:49 .= |[(|.|[x,y]|.| * ((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 - cn))),(|.|[x,y]|.| * (sqrt (1 - (((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 - cn)) ^2))))]| by A3, A31, Th51 .= |[s,r]| by A17, A29, A30, A32, A33 ; hence f . |[x,y]| = |[s,r]| by A3; ::_thesis: verum end; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) A34: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A34; hence ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then f2 is continuous by A3, A17, Th52; hence f is continuous by A6, A8, A27, A28, JGRAPH_2:35; ::_thesis: verum end; theorem Th57: :: JGRAPH_4:57 for cn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set sn = sqrt (1 - (cn ^2)); set p0 = |[cn,(sqrt (1 - (cn ^2)))]|; A1: |[cn,(sqrt (1 - (cn ^2)))]| `2 = sqrt (1 - (cn ^2)) by EUCLID:52; |[cn,(sqrt (1 - (cn ^2)))]| `1 = cn by EUCLID:52; then A2: |.|[cn,(sqrt (1 - (cn ^2)))]|.| = sqrt (((sqrt (1 - (cn ^2))) ^2) + (cn ^2)) by A1, JGRAPH_3:1; assume A3: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then cn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (cn ^2) > 0 by XREAL_1:50; then (sqrt (1 - (cn ^2))) ^2 = 1 - (cn ^2) by SQUARE_1:def_2; then A5: (|[cn,(sqrt (1 - (cn ^2)))]| `1) / |.|[cn,(sqrt (1 - (cn ^2)))]|.| = cn by A2, EUCLID:52, SQUARE_1:18; |[cn,(sqrt (1 - (cn ^2)))]| `2 > 0 by A1, A4, SQUARE_1:25; then A6: |[cn,(sqrt (1 - (cn ^2)))]| in K0 by A3, A5, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A7: rng (proj2 * ((cn -FanMorphN) | K1)) c= the carrier of R^1 by TOPMETR:17; A8: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & (p8 `1) / |.p8.| <= cn & p8 `2 >= 0 & p8 <> 0. (TOP-REAL 2) ) by A3; hence x in B0 by A3; ::_thesis: verum end; A9: dom ((cn -FanMorphN) | K1) c= dom (proj1 * ((cn -FanMorphN) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((cn -FanMorphN) | K1) or x in dom (proj1 * ((cn -FanMorphN) | K1)) ) assume A10: x in dom ((cn -FanMorphN) | K1) ; ::_thesis: x in dom (proj1 * ((cn -FanMorphN) | K1)) then x in (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61; then x in dom (cn -FanMorphN) by XBOOLE_0:def_4; then A11: ( dom proj1 = the carrier of (TOP-REAL 2) & (cn -FanMorphN) . x in rng (cn -FanMorphN) ) by FUNCT_1:3, FUNCT_2:def_1; ((cn -FanMorphN) | K1) . x = (cn -FanMorphN) . x by A10, FUNCT_1:47; hence x in dom (proj1 * ((cn -FanMorphN) | K1)) by A10, A11, FUNCT_1:11; ::_thesis: verum end; A12: rng (proj1 * ((cn -FanMorphN) | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1 * ((cn -FanMorphN) | K1)) c= dom ((cn -FanMorphN) | K1) by RELAT_1:25; then dom (proj1 * ((cn -FanMorphN) | K1)) = dom ((cn -FanMorphN) | K1) by A9, XBOOLE_0:def_10 .= (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g2 = proj1 * ((cn -FanMorphN) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A12, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) A13: dom ((cn -FanMorphN) | K1) = (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A14: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A15: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A14; then A16: (cn -FanMorphN) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))))]| by A3, Th51; ((cn -FanMorphN) | K1) . p = (cn -FanMorphN) . p by A15, A14, FUNCT_1:49; then g2 . p = proj1 . |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))))]| by A15, A13, A14, A16, FUNCT_1:13 .= |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))))]| `1 by PSCOMP_1:def_5 .= |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) by EUCLID:52 ; hence g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A17: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ; A18: dom ((cn -FanMorphN) | K1) c= dom (proj2 * ((cn -FanMorphN) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((cn -FanMorphN) | K1) or x in dom (proj2 * ((cn -FanMorphN) | K1)) ) assume A19: x in dom ((cn -FanMorphN) | K1) ; ::_thesis: x in dom (proj2 * ((cn -FanMorphN) | K1)) then x in (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61; then x in dom (cn -FanMorphN) by XBOOLE_0:def_4; then A20: ( dom proj2 = the carrier of (TOP-REAL 2) & (cn -FanMorphN) . x in rng (cn -FanMorphN) ) by FUNCT_1:3, FUNCT_2:def_1; ((cn -FanMorphN) | K1) . x = (cn -FanMorphN) . x by A19, FUNCT_1:47; hence x in dom (proj2 * ((cn -FanMorphN) | K1)) by A19, A20, FUNCT_1:11; ::_thesis: verum end; dom (proj2 * ((cn -FanMorphN) | K1)) c= dom ((cn -FanMorphN) | K1) by RELAT_1:25; then dom (proj2 * ((cn -FanMorphN) | K1)) = dom ((cn -FanMorphN) | K1) by A18, XBOOLE_0:def_10 .= (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g1 = proj2 * ((cn -FanMorphN) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A7, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) A21: dom ((cn -FanMorphN) | K1) = (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A22: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A23: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A22; then A24: (cn -FanMorphN) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))))]| by A3, Th51; ((cn -FanMorphN) | K1) . p = (cn -FanMorphN) . p by A23, A22, FUNCT_1:49; then g1 . p = proj2 . |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))))]| by A23, A21, A22, A24, FUNCT_1:13 .= |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))))]| `2 by PSCOMP_1:def_6 .= |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) by EUCLID:52 ; hence g1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A25: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) A26: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A26; hence ( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then A27: f1 is continuous by A3, A25, Th55; A28: for x, y, s, r being real number st |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| holds f . |[x,y]| = |[s,r]| proof let x, y, s, r be real number ; ::_thesis: ( |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| implies f . |[x,y]| = |[s,r]| ) assume that A29: |[x,y]| in K1 and A30: ( s = f2 . |[x,y]| & r = f1 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[s,r]| set p99 = |[x,y]|; A31: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A29; A32: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then A33: f1 . |[x,y]| = |.|[x,y]|.| * (sqrt (1 - (((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 + cn)) ^2))) by A25, A29; ((cn -FanMorphN) | K0) . |[x,y]| = (cn -FanMorphN) . |[x,y]| by A29, FUNCT_1:49 .= |[(|.|[x,y]|.| * ((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 + cn))),(|.|[x,y]|.| * (sqrt (1 - (((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 + cn)) ^2))))]| by A3, A31, Th51 .= |[s,r]| by A17, A29, A30, A32, A33 ; hence f . |[x,y]| = |[s,r]| by A3; ::_thesis: verum end; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) A34: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A34; hence ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then f2 is continuous by A3, A17, Th53; hence f is continuous by A6, A8, A27, A28, JGRAPH_2:35; ::_thesis: verum end; theorem Th58: :: JGRAPH_4:58 for cn being Real for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= cn * |.p.| & p `2 >= 0 ) } holds K03 is closed proof defpred S1[ Point of (TOP-REAL 2)] means $1 `2 >= 0 ; let sn be Real; ::_thesis: for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= sn * |.p.| & p `2 >= 0 ) } holds K03 is closed let K003 be Subset of (TOP-REAL 2); ::_thesis: ( K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= sn * |.p.| & p `2 >= 0 ) } implies K003 is closed ) assume A1: K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= sn * |.p.| & p `2 >= 0 ) } ; ::_thesis: K003 is closed reconsider KX = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); defpred S2[ Point of (TOP-REAL 2)] means $1 `1 >= sn * |.$1.|; reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A2: { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ { p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } from DOMAIN_1:sch_10(); ( K1 is closed & KX is closed ) by Lm8, JORDAN6:7; hence K003 is closed by A1, A2, TOPS_1:8; ::_thesis: verum end; theorem Th59: :: JGRAPH_4:59 for cn being Real for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= cn * |.p.| & p `2 >= 0 ) } holds K03 is closed proof defpred S1[ Point of (TOP-REAL 2)] means $1 `2 >= 0 ; let sn be Real; ::_thesis: for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= sn * |.p.| & p `2 >= 0 ) } holds K03 is closed let K003 be Subset of (TOP-REAL 2); ::_thesis: ( K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= sn * |.p.| & p `2 >= 0 ) } implies K003 is closed ) assume A1: K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= sn * |.p.| & p `2 >= 0 ) } ; ::_thesis: K003 is closed reconsider KX = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); defpred S2[ Point of (TOP-REAL 2)] means $1 `1 <= sn * |.$1.|; reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A2: { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ { p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } from DOMAIN_1:sch_10(); ( K1 is closed & KX is closed ) by Lm10, JORDAN6:7; hence K003 is closed by A1, A2, TOPS_1:8; ::_thesis: verum end; theorem Th60: :: JGRAPH_4:60 for cn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set sn = sqrt (1 - (cn ^2)); set p0 = |[cn,(sqrt (1 - (cn ^2)))]|; A1: |[cn,(sqrt (1 - (cn ^2)))]| `2 = sqrt (1 - (cn ^2)) by EUCLID:52; assume A2: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then cn ^2 < 1 ^2 by SQUARE_1:50; then A3: 1 - (cn ^2) > 0 by XREAL_1:50; then A4: |[cn,(sqrt (1 - (cn ^2)))]| `2 > 0 by A1, SQUARE_1:25; then |[cn,(sqrt (1 - (cn ^2)))]| in K0 by A2, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; |[cn,(sqrt (1 - (cn ^2)))]| <> 0. (TOP-REAL 2) by A1, A3, JGRAPH_2:3, SQUARE_1:25; then not |[cn,(sqrt (1 - (cn ^2)))]| in {(0. (TOP-REAL 2))} by TARSKI:def_1; then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A2, XBOOLE_0:def_5; A5: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; |[cn,(sqrt (1 - (cn ^2)))]| `1 = cn by EUCLID:52; then A6: |.|[cn,(sqrt (1 - (cn ^2)))]|.| = sqrt (((sqrt (1 - (cn ^2))) ^2) + (cn ^2)) by A1, JGRAPH_3:1; A7: D <> {} ; (sqrt (1 - (cn ^2))) ^2 = 1 - (cn ^2) by A3, SQUARE_1:def_2; then A8: (|[cn,(sqrt (1 - (cn ^2)))]| `1) / |.|[cn,(sqrt (1 - (cn ^2)))]|.| = cn by A6, EUCLID:52, SQUARE_1:18; then A9: |[cn,(sqrt (1 - (cn ^2)))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } by A4, JGRAPH_2:3; A10: { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } c= K1 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) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 ) assume x in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in K1 then ex p being Point of (TOP-REAL 2) st ( p = x & (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) ; hence x in K1 by A2; ::_thesis: verum end; A11: { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } c= K1 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) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 ) assume x in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in K1 then ex p being Point of (TOP-REAL 2) st ( p = x & (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) ; hence x in K1 by A2; ::_thesis: verum end; then reconsider K00 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A9, PRE_TOPC:8; the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; then A12: rng (f | K00) c= D ; |[cn,(sqrt (1 - (cn ^2)))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } by A4, A8, JGRAPH_2:3; then reconsider K11 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A10, PRE_TOPC:8; the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; then A13: rng (f | K11) c= D ; the carrier of ((TOP-REAL 2) | B0) = the carrier of ((TOP-REAL 2) | D) ; then A14: dom f = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1 .= K1 by PRE_TOPC:8 ; then dom (f | K00) = K00 by A11, RELAT_1:62 .= the carrier of (((TOP-REAL 2) | K1) | K00) by PRE_TOPC:8 ; then reconsider f1 = f | K00 as Function of (((TOP-REAL 2) | K1) | K00),((TOP-REAL 2) | D) by A12, FUNCT_2:2; dom (f | K11) = K11 by A10, A14, RELAT_1:62 .= the carrier of (((TOP-REAL 2) | K1) | K11) by PRE_TOPC:8 ; then reconsider f2 = f | K11 as Function of (((TOP-REAL 2) | K1) | K11),((TOP-REAL 2) | D) by A13, FUNCT_2:2; defpred S1[ Point of (TOP-REAL 2)] means ( ($1 `1) / |.$1.| >= cn & $1 `2 >= 0 & $1 <> 0. (TOP-REAL 2) ); A15: dom f2 = the carrier of (((TOP-REAL 2) | K1) | K11) by FUNCT_2:def_1 .= K11 by PRE_TOPC:8 ; { p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K001 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A9; A16: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; defpred S2[ Point of (TOP-REAL 2)] means ( $1 `1 >= cn * |.$1.| & $1 `2 >= 0 ); { p where p is Point of (TOP-REAL 2) : S2[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= cn * |.p.| & p `2 >= 0 ) } as Subset of (TOP-REAL 2) ; defpred S3[ Point of (TOP-REAL 2)] means ( ($1 `1) / |.$1.| <= cn & $1 `2 >= 0 & $1 <> 0. (TOP-REAL 2) ); A17: { p where p is Point of (TOP-REAL 2) : S3[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); A18: rng ((cn -FanMorphN) | K001) c= K1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((cn -FanMorphN) | K001) or y in K1 ) assume y in rng ((cn -FanMorphN) | K001) ; ::_thesis: y in K1 then consider x being set such that A19: x in dom ((cn -FanMorphN) | K001) and A20: y = ((cn -FanMorphN) | K001) . x by FUNCT_1:def_3; x in dom (cn -FanMorphN) by A19, RELAT_1:57; then reconsider q = x as Point of (TOP-REAL 2) ; A21: y = (cn -FanMorphN) . q by A19, A20, FUNCT_1:47; dom ((cn -FanMorphN) | K001) = (dom (cn -FanMorphN)) /\ K001 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K001 by FUNCT_2:def_1 .= K001 by XBOOLE_1:28 ; then A22: ex p2 being Point of (TOP-REAL 2) st ( p2 = q & (p2 `1) / |.p2.| >= cn & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) by A19; then A23: ((q `1) / |.q.|) - cn >= 0 by XREAL_1:48; |.q.| <> 0 by A22, TOPRNS_1:24; then A24: |.q.| ^2 > 0 ^2 by SQUARE_1:12; set q4 = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]|; A25: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn)) by EUCLID:52; A26: 1 - cn > 0 by A2, XREAL_1:149; 0 <= (q `2) ^2 by XREAL_1:63; then 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then (q `1) ^2 <= |.q.| ^2 by JGRAPH_3:1; then ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A24, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `1) / |.q.| by SQUARE_1:51; then 1 - cn >= ((q `1) / |.q.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((q `1) / |.q.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A26, XREAL_1:72; then - 1 <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A26, XCMPLX_1:197; then ((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A26, A23, SQUARE_1:49; then A27: 1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A28: 1 - ((- ((((q `1) / |.q.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; sqrt (1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2)) >= 0 by A27, SQUARE_1:def_2; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) ^2) / ((1 - cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((q `1) / |.q.|) - cn) ^2) / ((1 - cn) ^2))) >= 0 ; then A29: sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) >= 0 by XCMPLX_1:76; A30: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) by EUCLID:52; then A31: (|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) by A28, SQUARE_1:def_2 ; |.|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]|.| ^2 = ((|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| `1) ^2) + ((|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A25, A31 ; then A32: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| <> 0. (TOP-REAL 2) by A24, TOPRNS_1:23; (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by A2, A22, Th51; hence y in K1 by A2, A21, A30, A29, A32; ::_thesis: verum end; A33: dom (cn -FanMorphN) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then dom ((cn -FanMorphN) | K001) = K001 by RELAT_1:62 .= the carrier of ((TOP-REAL 2) | K001) by PRE_TOPC:8 ; then reconsider f3 = (cn -FanMorphN) | K001 as Function of ((TOP-REAL 2) | K001),((TOP-REAL 2) | K1) by A5, A18, FUNCT_2:2; A34: K003 is closed by Th58; K1 c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in D ) assume A35: x in K1 ; ::_thesis: x in D then ex p6 being Point of (TOP-REAL 2) st ( p6 = x & p6 `2 >= 0 & p6 <> 0. (TOP-REAL 2) ) by A2; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in D by A2, A35, XBOOLE_0:def_5; ::_thesis: verum end; then D = K1 \/ D by XBOOLE_1:12; then A36: (TOP-REAL 2) | K1 is SubSpace of (TOP-REAL 2) | D by TOPMETR:4; |[cn,(sqrt (1 - (cn ^2)))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } by A4, A8, JGRAPH_2:3; then reconsider K111 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A17; A37: rng ((cn -FanMorphN) | K111) c= K1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((cn -FanMorphN) | K111) or y in K1 ) assume y in rng ((cn -FanMorphN) | K111) ; ::_thesis: y in K1 then consider x being set such that A38: x in dom ((cn -FanMorphN) | K111) and A39: y = ((cn -FanMorphN) | K111) . x by FUNCT_1:def_3; x in dom (cn -FanMorphN) by A38, RELAT_1:57; then reconsider q = x as Point of (TOP-REAL 2) ; A40: y = (cn -FanMorphN) . q by A38, A39, FUNCT_1:47; dom ((cn -FanMorphN) | K111) = (dom (cn -FanMorphN)) /\ K111 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K111 by FUNCT_2:def_1 .= K111 by XBOOLE_1:28 ; then A41: ex p2 being Point of (TOP-REAL 2) st ( p2 = q & (p2 `1) / |.p2.| <= cn & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) by A38; then A42: ((q `1) / |.q.|) - cn <= 0 by XREAL_1:47; |.q.| <> 0 by A41, TOPRNS_1:24; then A43: |.q.| ^2 > 0 ^2 by SQUARE_1:12; set q4 = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]|; A44: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) by EUCLID:52; A45: 1 + cn > 0 by A2, XREAL_1:148; 0 <= (q `2) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A43, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `1) / |.q.| by SQUARE_1:51; then (- 1) - cn <= ((q `1) / |.q.|) - cn by XREAL_1:9; then (- (1 + cn)) / (1 + cn) <= (((q `1) / |.q.|) - cn) / (1 + cn) by A45, XREAL_1:72; then - 1 <= (((q `1) / |.q.|) - cn) / (1 + cn) by A45, XCMPLX_1:197; then A46: ((((q `1) / |.q.|) - cn) / (1 + cn)) ^2 <= 1 ^2 by A45, A42, SQUARE_1:49; then A47: 1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2) >= 0 by XREAL_1:48; 1 - ((- ((((q `1) / |.q.|) - cn) / (1 + cn))) ^2) >= 0 by A46, XREAL_1:48; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2) >= 0 by XCMPLX_1:187; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) ^2) / ((1 + cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((q `1) / |.q.|) - cn) ^2) / ((1 + cn) ^2))) >= 0 ; then A48: sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) >= 0 by XCMPLX_1:76; A49: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) by EUCLID:52; then A50: (|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) by A47, SQUARE_1:def_2 ; |.|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]|.| ^2 = ((|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| `1) ^2) + ((|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A44, A50 ; then A51: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| <> 0. (TOP-REAL 2) by A43, TOPRNS_1:23; (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| by A2, A41, Th51; hence y in K1 by A2, A40, A49, A48, A51; ::_thesis: verum end; dom ((cn -FanMorphN) | K111) = K111 by A33, RELAT_1:62 .= the carrier of ((TOP-REAL 2) | K111) by PRE_TOPC:8 ; then reconsider f4 = (cn -FanMorphN) | K111 as Function of ((TOP-REAL 2) | K111),((TOP-REAL 2) | K1) by A16, A37, FUNCT_2:2; the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then ( ((TOP-REAL 2) | K1) | K11 = (TOP-REAL 2) | K111 & f2 = f4 ) by A2, FUNCT_1:51, GOBOARD9:2; then A52: f2 is continuous by A2, A36, Th57, PRE_TOPC:26; A53: the carrier of ((TOP-REAL 2) | K1) = K0 by PRE_TOPC:8; set T1 = ((TOP-REAL 2) | K1) | K00; set T2 = ((TOP-REAL 2) | K1) | K11; A54: [#] (((TOP-REAL 2) | K1) | K11) = K11 by PRE_TOPC:def_5; defpred S4[ Point of (TOP-REAL 2)] means ( $1 `1 <= cn * |.$1.| & $1 `2 >= 0 ); { p where p is Point of (TOP-REAL 2) : S4[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K004 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= cn * |.p.| & p `2 >= 0 ) } as Subset of (TOP-REAL 2) ; A55: K004 /\ K1 c= K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K004 /\ K1 or x in K11 ) assume A56: x in K004 /\ K1 ; ::_thesis: x in K11 then x in K004 by XBOOLE_0:def_4; then consider q1 being Point of (TOP-REAL 2) such that A57: q1 = x and A58: q1 `1 <= cn * |.q1.| and q1 `2 >= 0 ; x in K1 by A56, XBOOLE_0:def_4; then A59: ex q2 being Point of (TOP-REAL 2) st ( q2 = x & q2 `2 >= 0 & q2 <> 0. (TOP-REAL 2) ) by A2; (q1 `1) / |.q1.| <= (cn * |.q1.|) / |.q1.| by A58, XREAL_1:72; then (q1 `1) / |.q1.| <= cn by A57, A59, TOPRNS_1:24, XCMPLX_1:89; hence x in K11 by A57, A59; ::_thesis: verum end; A60: K004 is closed by Th59; the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then ( ((TOP-REAL 2) | K1) | K00 = (TOP-REAL 2) | K001 & f1 = f3 ) by A2, FUNCT_1:51, GOBOARD9:2; then A61: f1 is continuous by A2, A36, Th56, PRE_TOPC:26; A62: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def_5; K11 c= K004 /\ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K11 or x in K004 /\ K1 ) assume x in K11 ; ::_thesis: x in K004 /\ K1 then consider p being Point of (TOP-REAL 2) such that A63: p = x and A64: (p `1) / |.p.| <= cn and A65: p `2 >= 0 and A66: p <> 0. (TOP-REAL 2) ; ((p `1) / |.p.|) * |.p.| <= cn * |.p.| by A64, XREAL_1:64; then p `1 <= cn * |.p.| by A66, TOPRNS_1:24, XCMPLX_1:87; then A67: x in K004 by A63, A65; x in K1 by A2, A63, A65, A66; hence x in K004 /\ K1 by A67, XBOOLE_0:def_4; ::_thesis: verum end; then K11 = K004 /\ ([#] ((TOP-REAL 2) | K1)) by A62, A55, XBOOLE_0:def_10; then A68: K11 is closed by A60, PRE_TOPC:13; A69: K003 /\ K1 c= K00 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K003 /\ K1 or x in K00 ) assume A70: x in K003 /\ K1 ; ::_thesis: x in K00 then x in K003 by XBOOLE_0:def_4; then consider q1 being Point of (TOP-REAL 2) such that A71: q1 = x and A72: q1 `1 >= cn * |.q1.| and q1 `2 >= 0 ; x in K1 by A70, XBOOLE_0:def_4; then A73: ex q2 being Point of (TOP-REAL 2) st ( q2 = x & q2 `2 >= 0 & q2 <> 0. (TOP-REAL 2) ) by A2; (q1 `1) / |.q1.| >= (cn * |.q1.|) / |.q1.| by A72, XREAL_1:72; then (q1 `1) / |.q1.| >= cn by A71, A73, TOPRNS_1:24, XCMPLX_1:89; hence x in K00 by A71, A73; ::_thesis: verum end; K00 c= K003 /\ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K00 or x in K003 /\ K1 ) assume x in K00 ; ::_thesis: x in K003 /\ K1 then consider p being Point of (TOP-REAL 2) such that A74: p = x and A75: (p `1) / |.p.| >= cn and A76: p `2 >= 0 and A77: p <> 0. (TOP-REAL 2) ; ((p `1) / |.p.|) * |.p.| >= cn * |.p.| by A75, XREAL_1:64; then p `1 >= cn * |.p.| by A77, TOPRNS_1:24, XCMPLX_1:87; then A78: x in K003 by A74, A76; x in K1 by A2, A74, A76, A77; hence x in K003 /\ K1 by A78, XBOOLE_0:def_4; ::_thesis: verum end; then K00 = K003 /\ ([#] ((TOP-REAL 2) | K1)) by A62, A69, XBOOLE_0:def_10; then A79: K00 is closed by A34, PRE_TOPC:13; A80: [#] (((TOP-REAL 2) | K1) | K00) = K00 by PRE_TOPC:def_5; A81: for p being set st p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) holds f1 . p = f2 . p proof let p be set ; ::_thesis: ( p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) implies f1 . p = f2 . p ) assume A82: p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) ; ::_thesis: f1 . p = f2 . p then p in K00 by A80, XBOOLE_0:def_4; hence f1 . p = f . p by FUNCT_1:49 .= f2 . p by A54, A82, FUNCT_1:49 ; ::_thesis: verum end; A83: K1 c= K00 \/ K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in K00 \/ K11 ) assume x in K1 ; ::_thesis: x in K00 \/ K11 then consider p being Point of (TOP-REAL 2) such that A84: ( p = x & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) by A2; percases ( (p `1) / |.p.| >= cn or (p `1) / |.p.| < cn ) ; suppose (p `1) / |.p.| >= cn ; ::_thesis: x in K00 \/ K11 then x in K00 by A84; hence x in K00 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; suppose (p `1) / |.p.| < cn ; ::_thesis: x in K00 \/ K11 then x in K11 by A84; hence x in K00 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; end; end; then ([#] (((TOP-REAL 2) | K1) | K00)) \/ ([#] (((TOP-REAL 2) | K1) | K11)) = [#] ((TOP-REAL 2) | K1) by A80, A54, A62, XBOOLE_0:def_10; then consider h being Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) such that A85: h = f1 +* f2 and A86: h is continuous by A80, A54, A79, A68, A61, A52, A81, JGRAPH_2:1; A87: dom h = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; A88: dom f1 = the carrier of (((TOP-REAL 2) | K1) | K00) by FUNCT_2:def_1 .= K00 by PRE_TOPC:8 ; A89: for y being set st y in dom h holds h . y = f . y proof let y be set ; ::_thesis: ( y in dom h implies h . y = f . y ) assume A90: y in dom h ; ::_thesis: h . y = f . y percases ( ( y in K00 & not y in K11 ) or y in K11 ) by A83, A87, A53, A90, XBOOLE_0:def_3; supposeA91: ( y in K00 & not y in K11 ) ; ::_thesis: h . y = f . y then y in (dom f1) \/ (dom f2) by A88, XBOOLE_0:def_3; hence h . y = f1 . y by A15, A85, A91, FUNCT_4:def_1 .= f . y by A91, FUNCT_1:49 ; ::_thesis: verum end; supposeA92: y in K11 ; ::_thesis: h . y = f . y then y in (dom f1) \/ (dom f2) by A15, XBOOLE_0:def_3; hence h . y = f2 . y by A15, A85, A92, FUNCT_4:def_1 .= f . y by A92, FUNCT_1:49 ; ::_thesis: verum end; end; end; K0 = the carrier of ((TOP-REAL 2) | K0) by PRE_TOPC:8 .= dom f by A7, FUNCT_2:def_1 ; hence f is continuous by A86, A87, A89, FUNCT_1:2, PRE_TOPC:8; ::_thesis: verum end; theorem Th61: :: JGRAPH_4:61 for cn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set sn = sqrt (1 - (cn ^2)); set p0 = |[cn,(- (sqrt (1 - (cn ^2))))]|; assume A1: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then cn ^2 < 1 ^2 by SQUARE_1:50; then 1 - (cn ^2) > 0 by XREAL_1:50; then ( |[cn,(- (sqrt (1 - (cn ^2))))]| `2 = - (sqrt (1 - (cn ^2))) & - (- (sqrt (1 - (cn ^2)))) > 0 ) by EUCLID:52, SQUARE_1:25; then A2: |[cn,(- (sqrt (1 - (cn ^2))))]| `2 < 0 ; then |[cn,(- (sqrt (1 - (cn ^2))))]| in K0 by A1, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; not |[cn,(- (sqrt (1 - (cn ^2))))]| in {(0. (TOP-REAL 2))} by A2, JGRAPH_2:3, TARSKI:def_1; then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A1, XBOOLE_0:def_5; A3: K1 c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in D ) assume x in K1 ; ::_thesis: x in D then consider p2 being Point of (TOP-REAL 2) such that A4: p2 = x and p2 `2 <= 0 and A5: p2 <> 0. (TOP-REAL 2) by A1; not p2 in {(0. (TOP-REAL 2))} by A5, TARSKI:def_1; hence x in D by A1, A4, XBOOLE_0:def_5; ::_thesis: verum end; for p being Point of ((TOP-REAL 2) | K1) for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) proof let p be Point of ((TOP-REAL 2) | K1); ::_thesis: for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) let V be Subset of ((TOP-REAL 2) | D); ::_thesis: ( f . p in V & V is open implies ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) ) assume that A6: f . p in V and A7: V is open ; ::_thesis: ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) consider V2 being Subset of (TOP-REAL 2) such that A8: V2 is open and A9: V2 /\ ([#] ((TOP-REAL 2) | D)) = V by A7, TOPS_2:24; reconsider W2 = V2 /\ ([#] ((TOP-REAL 2) | K1)) as Subset of ((TOP-REAL 2) | K1) ; A10: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def_5; then A11: f . p = (cn -FanMorphN) . p by A1, FUNCT_1:49; A12: f .: W2 c= V proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f .: W2 or y in V ) assume y in f .: W2 ; ::_thesis: y in V then consider x being set such that A13: x in dom f and A14: x in W2 and A15: y = f . x by FUNCT_1:def_6; f is Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) ; then dom f = K1 by A10, FUNCT_2:def_1; then consider p4 being Point of (TOP-REAL 2) such that A16: x = p4 and A17: p4 `2 <= 0 and p4 <> 0. (TOP-REAL 2) by A1, A13; A18: p4 in V2 by A14, A16, XBOOLE_0:def_4; p4 in [#] ((TOP-REAL 2) | K1) by A13, A16; then p4 in D by A3, A10; then A19: p4 in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; f . p4 = (cn -FanMorphN) . p4 by A1, A10, A13, A16, FUNCT_1:49 .= p4 by A17, Th49 ; hence y in V by A9, A15, A16, A18, A19, XBOOLE_0:def_4; ::_thesis: verum end; p in the carrier of ((TOP-REAL 2) | K1) ; then consider q being Point of (TOP-REAL 2) such that A20: q = p and A21: q `2 <= 0 and q <> 0. (TOP-REAL 2) by A1, A10; (cn -FanMorphN) . q = q by A21, Th49; then p in V2 by A6, A9, A11, A20, XBOOLE_0:def_4; then A22: p in W2 by XBOOLE_0:def_4; W2 is open by A8, TOPS_2:24; hence ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) by A22, A12; ::_thesis: verum end; hence f is continuous by JGRAPH_2:10; ::_thesis: verum end; theorem Th62: :: JGRAPH_4:62 for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds K0 is closed proof set J0 = NonZero (TOP-REAL 2); defpred S1[ Point of (TOP-REAL 2)] means $1 `2 >= 0 ; set I1 = { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } ; let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds K0 is closed let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies K0 is closed ) reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A1: { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch_2(); assume ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: K0 is closed then ( K1 is closed & K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) ) by A1, JORDAN6:7, PRE_TOPC:def_5; hence K0 is closed by PRE_TOPC:13; ::_thesis: verum end; theorem Th63: :: JGRAPH_4:63 for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds K0 is closed proof set J0 = NonZero (TOP-REAL 2); defpred S1[ Point of (TOP-REAL 2)] means $1 `2 <= 0 ; set I1 = { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } ; let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds K0 is closed let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies K0 is closed ) reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A1: { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch_2(); assume ( B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: K0 is closed then ( K1 is closed & K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) ) by A1, JORDAN6:8, PRE_TOPC:def_5; hence K0 is closed by PRE_TOPC:13; ::_thesis: verum end; theorem Th64: :: JGRAPH_4:64 for cn being Real for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:8; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & p8 `2 >= 0 & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; hence f is continuous by A1, Th60; ::_thesis: verum end; theorem Th65: :: JGRAPH_4:65 for cn being Real for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:8; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & p8 `2 <= 0 & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; hence f is continuous by A1, Th61; ::_thesis: verum end; theorem Th66: :: JGRAPH_4:66 for cn being Real for p being Point of (TOP-REAL 2) holds |.((cn -FanMorphN) . p).| = |.p.| proof let cn be Real; ::_thesis: for p being Point of (TOP-REAL 2) holds |.((cn -FanMorphN) . p).| = |.p.| let p be Point of (TOP-REAL 2); ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| set f = cn -FanMorphN ; set z = (cn -FanMorphN) . p; reconsider q = p as Point of (TOP-REAL 2) ; reconsider qz = (cn -FanMorphN) . p as Point of (TOP-REAL 2) ; percases ( ( (q `1) / |.q.| >= cn & q `2 > 0 ) or ( (q `1) / |.q.| < cn & q `2 > 0 ) or q `2 <= 0 ) ; supposeA1: ( (q `1) / |.q.| >= cn & q `2 > 0 ) ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| then A2: (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by Th49; then A3: qz `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) by EUCLID:52; A4: qz `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn)) by A2, EUCLID:52; A5: ((q `1) / |.q.|) - cn >= 0 by A1, XREAL_1:48; A6: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; |.q.| <> 0 by A1, JGRAPH_2:3, TOPRNS_1:24; then A7: |.q.| ^2 > 0 by SQUARE_1:12; 0 <= (q `2) ^2 by XREAL_1:63; then 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by A6, XREAL_1:72; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A7, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `1) / |.q.| by SQUARE_1:51; then A8: 1 - cn >= ((q `1) / |.q.|) - cn by XREAL_1:9; percases ( 1 - cn = 0 or 1 - cn <> 0 ) ; supposeA9: 1 - cn = 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| A10: (((q `1) / |.q.|) - cn) / (1 - cn) = (((q `1) / |.q.|) - cn) * ((1 - cn) ") by XCMPLX_0:def_9 .= (((q `1) / |.q.|) - cn) * 0 by A9 .= 0 ; then 1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2) = 1 ; then (cn -FanMorphN) . q = |[(|.q.| * 0),(|.q.| * 1)]| by A1, A10, Th49, SQUARE_1:18 .= |[0,|.q.|]| ; then ( ((cn -FanMorphN) . q) `2 = |.q.| & ((cn -FanMorphN) . q) `1 = 0 ) by EUCLID:52; then |.((cn -FanMorphN) . p).| = sqrt ((|.q.| ^2) + (0 ^2)) by JGRAPH_3:1 .= |.q.| by SQUARE_1:22 ; hence |.((cn -FanMorphN) . p).| = |.p.| ; ::_thesis: verum end; supposeA11: 1 - cn <> 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| percases ( 1 - cn > 0 or 1 - cn < 0 ) by A11; supposeA12: 1 - cn > 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| - (1 - cn) <= - (((q `1) / |.q.|) - cn) by A8, XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A12, XREAL_1:72; then - 1 <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A12, XCMPLX_1:197; then ((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A5, A12, SQUARE_1:49; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A13: 1 - ((- ((((q `1) / |.q.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; A14: (qz `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) ^2) by A3 .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) by A13, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A4, A14 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; hence |.((cn -FanMorphN) . p).| = |.p.| by SQUARE_1:22; ::_thesis: verum end; supposeA15: 1 - cn < 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| 0 + ((q `1) ^2) < ((q `1) ^2) + ((q `2) ^2) by A1, SQUARE_1:12, XREAL_1:8; then ((q `1) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A7, A6, XREAL_1:74; then ((q `1) ^2) / (|.q.| ^2) < 1 by A7, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 < 1 by XCMPLX_1:76; then A16: 1 > (q `1) / |.p.| by SQUARE_1:52; ((q `1) / |.q.|) - cn >= 0 by A1, XREAL_1:48; hence |.((cn -FanMorphN) . p).| = |.p.| by A15, A16, XREAL_1:9; ::_thesis: verum end; end; end; end; end; supposeA17: ( (q `1) / |.q.| < cn & q `2 > 0 ) ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| then |.q.| <> 0 by JGRAPH_2:3, TOPRNS_1:24; then A18: |.q.| ^2 > 0 by SQUARE_1:12; A19: ((q `1) / |.q.|) - cn < 0 by A17, XREAL_1:49; A20: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; 0 <= (q `2) ^2 by XREAL_1:63; then 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by A20, XREAL_1:72; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A18, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `1) / |.q.| by SQUARE_1:51; then A21: (- 1) - cn <= ((q `1) / |.q.|) - cn by XREAL_1:9; A22: (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| by A17, Th50; then A23: qz `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) by EUCLID:52; A24: qz `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) by A22, EUCLID:52; percases ( 1 + cn = 0 or 1 + cn <> 0 ) ; supposeA25: 1 + cn = 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| (((q `1) / |.q.|) - cn) / (1 + cn) = (((q `1) / |.q.|) - cn) * ((1 + cn) ") by XCMPLX_0:def_9 .= (((q `1) / |.q.|) - cn) * 0 by A25 .= 0 ; then ( ((cn -FanMorphN) . q) `2 = |.q.| & ((cn -FanMorphN) . q) `1 = 0 ) by A22, EUCLID:52, SQUARE_1:18; then |.((cn -FanMorphN) . p).| = sqrt ((|.q.| ^2) + (0 ^2)) by JGRAPH_3:1 .= |.q.| by SQUARE_1:22 ; hence |.((cn -FanMorphN) . p).| = |.p.| ; ::_thesis: verum end; supposeA26: 1 + cn <> 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| percases ( 1 + cn > 0 or 1 + cn < 0 ) by A26; supposeA27: 1 + cn > 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| then (- (1 + cn)) / (1 + cn) <= (((q `1) / |.q.|) - cn) / (1 + cn) by A21, XREAL_1:72; then - 1 <= (((q `1) / |.q.|) - cn) / (1 + cn) by A27, XCMPLX_1:197; then ((((q `1) / |.q.|) - cn) / (1 + cn)) ^2 <= 1 ^2 by A19, A27, SQUARE_1:49; then A28: 1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2) >= 0 by XREAL_1:48; A29: (qz `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) ^2) by A23 .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) by A28, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A24, A29 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; hence |.((cn -FanMorphN) . p).| = |.p.| by SQUARE_1:22; ::_thesis: verum end; supposeA30: 1 + cn < 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| 0 + ((q `1) ^2) < ((q `1) ^2) + ((q `2) ^2) by A17, SQUARE_1:12, XREAL_1:8; then ((q `1) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A18, A20, XREAL_1:74; then ((q `1) ^2) / (|.q.| ^2) < 1 by A18, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (q `1) / |.p.| by SQUARE_1:52; then A31: ((q `1) / |.q.|) - cn > (- 1) - cn by XREAL_1:9; - (1 + cn) > - 0 by A30, XREAL_1:24; hence |.((cn -FanMorphN) . p).| = |.p.| by A17, A31, XREAL_1:49; ::_thesis: verum end; end; end; end; end; suppose q `2 <= 0 ; ::_thesis: |.((cn -FanMorphN) . p).| = |.p.| hence |.((cn -FanMorphN) . p).| = |.p.| by Th49; ::_thesis: verum end; end; end; theorem Th67: :: JGRAPH_4:67 for cn being Real for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds (cn -FanMorphN) . x in K0 proof let cn be Real; ::_thesis: for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds (cn -FanMorphN) . x in K0 let x, K0 be set ; ::_thesis: ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies (cn -FanMorphN) . x in K0 ) assume A1: ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: (cn -FanMorphN) . x in K0 then consider p being Point of (TOP-REAL 2) such that A2: p = x and A3: p `2 >= 0 and A4: p <> 0. (TOP-REAL 2) ; A5: now__::_thesis:_not_|.p.|_<=_0 assume |.p.| <= 0 ; ::_thesis: contradiction then |.p.| = 0 ; hence contradiction by A4, TOPRNS_1:24; ::_thesis: verum end; then A6: |.p.| ^2 > 0 by SQUARE_1:12; percases ( (p `1) / |.p.| <= cn or (p `1) / |.p.| > cn ) ; supposeA7: (p `1) / |.p.| <= cn ; ::_thesis: (cn -FanMorphN) . x in K0 reconsider p9 = (cn -FanMorphN) . p as Point of (TOP-REAL 2) ; (cn -FanMorphN) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))))]| by A1, A3, A4, A7, Th51; then A8: p9 `2 = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) by EUCLID:52; A9: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; A10: 1 + cn > 0 by A1, XREAL_1:148; percases ( p `2 = 0 or p `2 <> 0 ) ; suppose p `2 = 0 ; ::_thesis: (cn -FanMorphN) . x in K0 hence (cn -FanMorphN) . x in K0 by A1, A2, Th49; ::_thesis: verum end; suppose p `2 <> 0 ; ::_thesis: (cn -FanMorphN) . x in K0 then 0 + ((p `1) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8; then ((p `1) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A9, XREAL_1:74; then ((p `1) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60; then ((p `1) / |.p.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (p `1) / |.p.| by SQUARE_1:52; then (- 1) - cn < ((p `1) / |.p.|) - cn by XREAL_1:9; then ((- 1) * (1 + cn)) / (1 + cn) < (((p `1) / |.p.|) - cn) / (1 + cn) by A10, XREAL_1:74; then A11: - 1 < (((p `1) / |.p.|) - cn) / (1 + cn) by A10, XCMPLX_1:89; ((p `1) / |.p.|) - cn <= 0 by A7, XREAL_1:47; then 1 ^2 > ((((p `1) / |.p.|) - cn) / (1 + cn)) ^2 by A10, A11, SQUARE_1:50; then 1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)) > 0 by SQUARE_1:25; then |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) > 0 by A5, XREAL_1:129; hence (cn -FanMorphN) . x in K0 by A1, A2, A8, JGRAPH_2:3; ::_thesis: verum end; end; end; supposeA12: (p `1) / |.p.| > cn ; ::_thesis: (cn -FanMorphN) . x in K0 reconsider p9 = (cn -FanMorphN) . p as Point of (TOP-REAL 2) ; (cn -FanMorphN) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))))]| by A1, A3, A4, A12, Th51; then A13: p9 `2 = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) by EUCLID:52; A14: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; A15: 1 - cn > 0 by A1, XREAL_1:149; percases ( p `2 = 0 or p `2 <> 0 ) ; suppose p `2 = 0 ; ::_thesis: (cn -FanMorphN) . x in K0 hence (cn -FanMorphN) . x in K0 by A1, A2, Th49; ::_thesis: verum end; suppose p `2 <> 0 ; ::_thesis: (cn -FanMorphN) . x in K0 then 0 + ((p `1) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8; then ((p `1) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A14, XREAL_1:74; then ((p `1) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60; then ((p `1) / |.p.|) ^2 < 1 by XCMPLX_1:76; then (p `1) / |.p.| < 1 by SQUARE_1:52; then ((p `1) / |.p.|) - cn < 1 - cn by XREAL_1:9; then (((p `1) / |.p.|) - cn) / (1 - cn) < (1 - cn) / (1 - cn) by A15, XREAL_1:74; then A16: (((p `1) / |.p.|) - cn) / (1 - cn) < 1 by A15, XCMPLX_1:60; ( - (1 - cn) < - 0 & ((p `1) / |.p.|) - cn >= cn - cn ) by A12, A15, XREAL_1:9, XREAL_1:24; then ((- 1) * (1 - cn)) / (1 - cn) < (((p `1) / |.p.|) - cn) / (1 - cn) by A15, XREAL_1:74; then - 1 < (((p `1) / |.p.|) - cn) / (1 - cn) by A15, XCMPLX_1:89; then 1 ^2 > ((((p `1) / |.p.|) - cn) / (1 - cn)) ^2 by A16, SQUARE_1:50; then 1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)) > 0 by SQUARE_1:25; then p9 `2 > 0 by A5, A13, XREAL_1:129; hence (cn -FanMorphN) . x in K0 by A1, A2, JGRAPH_2:3; ::_thesis: verum end; end; end; end; end; theorem Th68: :: JGRAPH_4:68 for cn being Real for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds (cn -FanMorphN) . x in K0 proof let cn be Real; ::_thesis: for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds (cn -FanMorphN) . x in K0 let x, K0 be set ; ::_thesis: ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies (cn -FanMorphN) . x in K0 ) assume A1: ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: (cn -FanMorphN) . x in K0 then ex p being Point of (TOP-REAL 2) st ( p = x & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) ; hence (cn -FanMorphN) . x in K0 by A1, Th49; ::_thesis: verum end; theorem Th69: :: JGRAPH_4:69 for cn being Real for D being non empty Subset of (TOP-REAL 2) st - 1 < cn & cn < 1 & D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphN) | D & h is continuous ) proof |[0,1]| `2 = 1 by EUCLID:52; then A1: |[0,1]| in { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } by JGRAPH_2:3; set Y1 = |[0,(- 1)]|; reconsider B0 = {(0. (TOP-REAL 2))} as Subset of (TOP-REAL 2) ; defpred S1[ Point of (TOP-REAL 2)] means $1 `2 >= 0 ; let cn be Real; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st - 1 < cn & cn < 1 & D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphN) | D & h is continuous ) let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & D ` = {(0. (TOP-REAL 2))} implies ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphN) | D & h is continuous ) ) assume that A2: ( - 1 < cn & cn < 1 ) and A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphN) | D & h is continuous ) A4: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; A5: D = B0 ` by A3 .= NonZero (TOP-REAL 2) by SUBSET_1:def_4 ; { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) from JGRAPH_4:sch_1(A5); then reconsider K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A1; A6: K0 = the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; A7: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; A8: rng ((cn -FanMorphN) | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((cn -FanMorphN) | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) ) assume y in rng ((cn -FanMorphN) | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0) then consider x being set such that A9: x in dom ((cn -FanMorphN) | K0) and A10: y = ((cn -FanMorphN) | K0) . x by FUNCT_1:def_3; x in (dom (cn -FanMorphN)) /\ K0 by A9, RELAT_1:61; then A11: x in K0 by XBOOLE_0:def_4; K0 c= the carrier of (TOP-REAL 2) by A7, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A11; (cn -FanMorphN) . p = y by A10, A11, FUNCT_1:49; then y in K0 by A2, A11, Th67; hence y in the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; ::_thesis: verum end; A12: K0 c= the carrier of (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K0 or z in the carrier of (TOP-REAL 2) ) assume z in K0 ; ::_thesis: z in the carrier of (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & p8 `2 >= 0 & p8 <> 0. (TOP-REAL 2) ) ; hence z in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; ( |[0,(- 1)]| `2 = - 1 & 0. (TOP-REAL 2) <> |[0,(- 1)]| ) by EUCLID:52, JGRAPH_2:3; then A13: |[0,(- 1)]| in { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ; A14: the carrier of ((TOP-REAL 2) | D) = NonZero (TOP-REAL 2) by A5, PRE_TOPC:8; defpred S2[ Point of (TOP-REAL 2)] means $1 `2 <= 0 ; { p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) from JGRAPH_4:sch_1(A5); then reconsider K1 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A13; A15: ( K0 is closed & K1 is closed ) by A5, Th62, Th63; dom ((cn -FanMorphN) | K0) = (dom (cn -FanMorphN)) /\ K0 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K0 by FUNCT_2:def_1 .= K0 by A12, XBOOLE_1:28 ; then reconsider f = (cn -FanMorphN) | K0 as Function of (((TOP-REAL 2) | D) | K0),((TOP-REAL 2) | D) by A6, A8, FUNCT_2:2, XBOOLE_1:1; A16: K1 = the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; A17: rng ((cn -FanMorphN) | K1) c= the carrier of (((TOP-REAL 2) | D) | K1) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((cn -FanMorphN) | K1) or y in the carrier of (((TOP-REAL 2) | D) | K1) ) assume y in rng ((cn -FanMorphN) | K1) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K1) then consider x being set such that A18: x in dom ((cn -FanMorphN) | K1) and A19: y = ((cn -FanMorphN) | K1) . x by FUNCT_1:def_3; x in (dom (cn -FanMorphN)) /\ K1 by A18, RELAT_1:61; then A20: x in K1 by XBOOLE_0:def_4; K1 c= the carrier of (TOP-REAL 2) by A7, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A20; (cn -FanMorphN) . p = y by A19, A20, FUNCT_1:49; then y in K1 by A2, A20, Th68; hence y in the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; ::_thesis: verum end; A21: K1 c= the carrier of (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in the carrier of (TOP-REAL 2) ) assume z in K1 ; ::_thesis: z in the carrier of (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & p8 `2 <= 0 & p8 <> 0. (TOP-REAL 2) ) ; hence z in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; dom ((cn -FanMorphN) | K1) = (dom (cn -FanMorphN)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by A21, XBOOLE_1:28 ; then reconsider g = (cn -FanMorphN) | K1 as Function of (((TOP-REAL 2) | D) | K1),((TOP-REAL 2) | D) by A16, A17, FUNCT_2:2, XBOOLE_1:1; A22: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5; A23: D c= K0 \/ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in K0 \/ K1 ) assume A24: x in D ; ::_thesis: x in K0 \/ K1 then reconsider px = x as Point of (TOP-REAL 2) ; not x in {(0. (TOP-REAL 2))} by A5, A24, XBOOLE_0:def_5; then ( ( px `2 >= 0 & px <> 0. (TOP-REAL 2) ) or ( px `2 <= 0 & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1; then ( x in K0 or x in K1 ) ; hence x in K0 \/ K1 by XBOOLE_0:def_3; ::_thesis: verum end; A25: dom f = K0 by A6, FUNCT_2:def_1; A26: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5; A27: for x being set st x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) holds f . x = g . x proof let x be set ; ::_thesis: ( x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) implies f . x = g . x ) assume A28: x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) ; ::_thesis: f . x = g . x then x in K0 by A26, XBOOLE_0:def_4; then f . x = (cn -FanMorphN) . x by FUNCT_1:49; hence f . x = g . x by A22, A28, FUNCT_1:49; ::_thesis: verum end; D = [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; then A29: ([#] (((TOP-REAL 2) | D) | K0)) \/ ([#] (((TOP-REAL 2) | D) | K1)) = [#] ((TOP-REAL 2) | D) by A26, A22, A23, XBOOLE_0:def_10; A30: ( f is continuous & g is continuous ) by A2, A5, Th64, Th65; then consider h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) such that A31: h = f +* g and h is continuous by A26, A22, A29, A15, A27, JGRAPH_2:1; A32: dom h = the carrier of ((TOP-REAL 2) | D) by FUNCT_2:def_1; A33: dom g = K1 by A16, FUNCT_2:def_1; ( K0 = [#] (((TOP-REAL 2) | D) | K0) & K1 = [#] (((TOP-REAL 2) | D) | K1) ) by PRE_TOPC:def_5; then A34: f tolerates g by A27, A25, A33, PARTFUN1:def_4; A35: for x being set st x in dom h holds h . x = ((cn -FanMorphN) | D) . x proof let x be set ; ::_thesis: ( x in dom h implies h . x = ((cn -FanMorphN) | D) . x ) assume A36: x in dom h ; ::_thesis: h . x = ((cn -FanMorphN) | D) . x then reconsider p = x as Point of (TOP-REAL 2) by A14, XBOOLE_0:def_5; not x in {(0. (TOP-REAL 2))} by A14, A36, XBOOLE_0:def_5; then A37: x <> 0. (TOP-REAL 2) by TARSKI:def_1; A38: x in (D `) ` by A32, A36, PRE_TOPC:8; now__::_thesis:_(_(_x_in_K0_&_h_._x_=_((cn_-FanMorphN)_|_D)_._x_)_or_(_not_x_in_K0_&_h_._x_=_((cn_-FanMorphN)_|_D)_._x_)_) percases ( x in K0 or not x in K0 ) ; caseA39: x in K0 ; ::_thesis: h . x = ((cn -FanMorphN) | D) . x A40: ((cn -FanMorphN) | D) . p = (cn -FanMorphN) . p by A38, FUNCT_1:49 .= f . p by A39, FUNCT_1:49 ; h . p = (g +* f) . p by A31, A34, FUNCT_4:34 .= f . p by A25, A39, FUNCT_4:13 ; hence h . x = ((cn -FanMorphN) | D) . x by A40; ::_thesis: verum end; case not x in K0 ; ::_thesis: h . x = ((cn -FanMorphN) | D) . x then not p `2 >= 0 by A37; then A41: x in K1 by A37; ((cn -FanMorphN) | D) . p = (cn -FanMorphN) . p by A38, FUNCT_1:49 .= g . p by A41, FUNCT_1:49 ; hence h . x = ((cn -FanMorphN) | D) . x by A31, A33, A41, FUNCT_4:13; ::_thesis: verum end; end; end; hence h . x = ((cn -FanMorphN) | D) . x ; ::_thesis: verum end; dom (cn -FanMorphN) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then dom ((cn -FanMorphN) | D) = the carrier of (TOP-REAL 2) /\ D by RELAT_1:61 .= the carrier of ((TOP-REAL 2) | D) by A4, XBOOLE_1:28 ; then f +* g = (cn -FanMorphN) | D by A31, A32, A35, FUNCT_1:2; hence ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphN) | D & h is continuous ) by A26, A22, A29, A30, A15, A27, JGRAPH_2:1; ::_thesis: verum end; theorem Th70: :: JGRAPH_4:70 for cn being Real st - 1 < cn & cn < 1 holds ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = cn -FanMorphN & h is continuous ) proof reconsider D = NonZero (TOP-REAL 2) as non empty Subset of (TOP-REAL 2) by JGRAPH_2:9; let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = cn -FanMorphN & h is continuous ) ) assume that A1: - 1 < cn and A2: cn < 1 ; ::_thesis: ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = cn -FanMorphN & h is continuous ) reconsider f = cn -FanMorphN as Function of (TOP-REAL 2),(TOP-REAL 2) ; A3: f . (0. (TOP-REAL 2)) = 0. (TOP-REAL 2) by Th49, JGRAPH_2:3; A4: for p being Point of ((TOP-REAL 2) | D) holds f . p <> f . (0. (TOP-REAL 2)) proof let p be Point of ((TOP-REAL 2) | D); ::_thesis: f . p <> f . (0. (TOP-REAL 2)) A5: [#] ((TOP-REAL 2) | D) = D by PRE_TOPC:def_5; then reconsider q = p as Point of (TOP-REAL 2) by XBOOLE_0:def_5; not p in {(0. (TOP-REAL 2))} by A5, XBOOLE_0:def_5; then A6: not p = 0. (TOP-REAL 2) by TARSKI:def_1; now__::_thesis:_(_(_(q_`1)_/_|.q.|_>=_cn_&_q_`2_>=_0_&_f_._p_<>_f_._(0._(TOP-REAL_2))_)_or_(_(q_`1)_/_|.q.|_<_cn_&_q_`2_>=_0_&_f_._p_<>_f_._(0._(TOP-REAL_2))_)_or_(_q_`2_<_0_&_f_._p_<>_f_._(0._(TOP-REAL_2))_)_) percases ( ( (q `1) / |.q.| >= cn & q `2 >= 0 ) or ( (q `1) / |.q.| < cn & q `2 >= 0 ) or q `2 < 0 ) ; caseA7: ( (q `1) / |.q.| >= cn & q `2 >= 0 ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) set q9 = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]|; A8: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn)) by EUCLID:52; now__::_thesis:_not_|[(|.q.|_*_((((q_`1)_/_|.q.|)_-_cn)_/_(1_-_cn))),(|.q.|_*_(sqrt_(1_-_(((((q_`1)_/_|.q.|)_-_cn)_/_(1_-_cn))_^2))))]|_=_0._(TOP-REAL_2) assume A9: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction A10: |.q.| <> 0 ^2 by A6, TOPRNS_1:24; then sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) = sqrt (1 - 0) by A8, A9, JGRAPH_2:3, XCMPLX_1:6 .= 1 by SQUARE_1:18 ; hence contradiction by A9, A10, EUCLID:52, JGRAPH_2:3; ::_thesis: verum end; hence f . p <> f . (0. (TOP-REAL 2)) by A1, A2, A3, A6, A7, Th51; ::_thesis: verum end; caseA11: ( (q `1) / |.q.| < cn & q `2 >= 0 ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) set q9 = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]|; A12: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) by EUCLID:52; now__::_thesis:_not_|[(|.q.|_*_((((q_`1)_/_|.q.|)_-_cn)_/_(1_+_cn))),(|.q.|_*_(sqrt_(1_-_(((((q_`1)_/_|.q.|)_-_cn)_/_(1_+_cn))_^2))))]|_=_0._(TOP-REAL_2) assume A13: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction A14: |.q.| <> 0 ^2 by A6, TOPRNS_1:24; then sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) = sqrt (1 - 0) by A12, A13, JGRAPH_2:3, XCMPLX_1:6 .= 1 by SQUARE_1:18 ; hence contradiction by A13, A14, EUCLID:52, JGRAPH_2:3; ::_thesis: verum end; hence f . p <> f . (0. (TOP-REAL 2)) by A1, A2, A3, A6, A11, Th51; ::_thesis: verum end; case q `2 < 0 ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) then f . p = p by Th49; hence f . p <> f . (0. (TOP-REAL 2)) by A6, Th49, JGRAPH_2:3; ::_thesis: verum end; end; end; hence f . p <> f . (0. (TOP-REAL 2)) ; ::_thesis: verum end; A15: for V being Subset of (TOP-REAL 2) st f . (0. (TOP-REAL 2)) in V & V is open holds ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) proof reconsider u0 = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:67; let V be Subset of (TOP-REAL 2); ::_thesis: ( f . (0. (TOP-REAL 2)) in V & V is open implies ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) ) reconsider VV = V as Subset of (TopSpaceMetr (Euclid 2)) by Lm11; assume that A16: f . (0. (TOP-REAL 2)) in V and A17: V is open ; ::_thesis: ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) VV is open by A17, Lm11, PRE_TOPC:30; then consider r being real number such that A18: r > 0 and A19: Ball (u0,r) c= V by A3, A16, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider W1 = Ball (u0,r) as Subset of (TOP-REAL 2) ; A20: W1 is open by GOBOARD6:3; A21: f .: W1 c= W1 proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f .: W1 or z in W1 ) assume z in f .: W1 ; ::_thesis: z in W1 then consider y being set such that A22: y in dom f and A23: y in W1 and A24: z = f . y by FUNCT_1:def_6; z in rng f by A22, A24, FUNCT_1:def_3; then reconsider qz = z as Point of (TOP-REAL 2) ; reconsider q = y as Point of (TOP-REAL 2) by A22; reconsider qy = q as Point of (Euclid 2) by EUCLID:67; reconsider pz = qz as Point of (Euclid 2) by EUCLID:67; dist (u0,qy) < r by A23, METRIC_1:11; then A25: |.((0. (TOP-REAL 2)) - q).| < r by JGRAPH_1:28; now__::_thesis:_(_(_q_`2_<=_0_&_z_in_W1_)_or_(_q_<>_0._(TOP-REAL_2)_&_(q_`1)_/_|.q.|_>=_cn_&_q_`2_>=_0_&_z_in_W1_)_or_(_q_<>_0._(TOP-REAL_2)_&_(q_`1)_/_|.q.|_<_cn_&_q_`2_>=_0_&_z_in_W1_)_) percases ( q `2 <= 0 or ( q <> 0. (TOP-REAL 2) & (q `1) / |.q.| >= cn & q `2 >= 0 ) or ( q <> 0. (TOP-REAL 2) & (q `1) / |.q.| < cn & q `2 >= 0 ) ) by JGRAPH_2:3; case q `2 <= 0 ; ::_thesis: z in W1 hence z in W1 by A23, A24, Th49; ::_thesis: verum end; caseA26: ( q <> 0. (TOP-REAL 2) & (q `1) / |.q.| >= cn & q `2 >= 0 ) ; ::_thesis: z in W1 then A27: ((q `1) / |.q.|) - cn >= 0 by XREAL_1:48; 0 <= (q `2) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A28: ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; A29: 1 - cn > 0 by A2, XREAL_1:149; |.q.| <> 0 by A26, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A28, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `1) / |.q.| by SQUARE_1:51; then 1 - cn >= ((q `1) / |.q.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((q `1) / |.q.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A29, XREAL_1:72; then - 1 <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A29, XCMPLX_1:197; then ((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A29, A27, SQUARE_1:49; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A30: 1 - ((- ((((q `1) / |.q.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; A31: (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by A1, A2, A26, Th51; then A32: qz `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn)) by A24, EUCLID:52; qz `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) by A24, A31, EUCLID:52; then A33: (qz `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) by A30, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A32, A33 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; then A34: |.qz.| = |.q.| by SQUARE_1:22; |.(- q).| < r by A25, EUCLID:27; then |.q.| < r by TOPRNS_1:26; then |.(- qz).| < r by A34, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - qz).| < r by EUCLID:27; then dist (u0,pz) < r by JGRAPH_1:28; hence z in W1 by METRIC_1:11; ::_thesis: verum end; caseA35: ( q <> 0. (TOP-REAL 2) & (q `1) / |.q.| < cn & q `2 >= 0 ) ; ::_thesis: z in W1 0 <= (q `2) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A36: ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; A37: 1 + cn > 0 by A1, XREAL_1:148; |.q.| <> 0 by A35, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A36, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `1) / |.q.| by SQUARE_1:51; then - (- 1) >= - ((q `1) / |.q.|) by XREAL_1:24; then 1 + cn >= (- ((q `1) / |.q.|)) + cn by XREAL_1:7; then A38: (- (((q `1) / |.q.|) - cn)) / (1 + cn) <= 1 by A37, XREAL_1:185; cn - ((q `1) / |.q.|) >= 0 by A35, XREAL_1:48; then - 1 <= (- (((q `1) / |.q.|) - cn)) / (1 + cn) by A37; then ((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A38, SQUARE_1:49; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then A39: 1 - ((- ((((q `1) / |.q.|) - cn) / (1 + cn))) ^2) >= 0 by XCMPLX_1:187; A40: (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| by A1, A2, A35, Th51; then A41: qz `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) by A24, EUCLID:52; qz `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) by A24, A40, EUCLID:52; then A42: (qz `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) by A39, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A41, A42 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; then A43: |.qz.| = |.q.| by SQUARE_1:22; |.(- q).| < r by A25, EUCLID:27; then |.q.| < r by TOPRNS_1:26; then |.(- qz).| < r by A43, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - qz).| < r by EUCLID:27; then dist (u0,pz) < r by JGRAPH_1:28; hence z in W1 by METRIC_1:11; ::_thesis: verum end; end; end; hence z in W1 ; ::_thesis: verum end; u0 in W1 by A18, GOBOARD6:1; hence ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) by A19, A20, A21, XBOOLE_1:1; ::_thesis: verum end; A44: D ` = {(0. (TOP-REAL 2))} by JGRAPH_3:20; then ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphN) | D & h is continuous ) by A1, A2, Th69; hence ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = cn -FanMorphN & h is continuous ) by A3, A44, A4, A15, JGRAPH_3:3; ::_thesis: verum end; theorem Th71: :: JGRAPH_4:71 for cn being Real st - 1 < cn & cn < 1 holds cn -FanMorphN is one-to-one proof let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies cn -FanMorphN is one-to-one ) assume that A1: - 1 < cn and A2: cn < 1 ; ::_thesis: cn -FanMorphN is one-to-one for x1, x2 being set st x1 in dom (cn -FanMorphN) & x2 in dom (cn -FanMorphN) & (cn -FanMorphN) . x1 = (cn -FanMorphN) . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom (cn -FanMorphN) & x2 in dom (cn -FanMorphN) & (cn -FanMorphN) . x1 = (cn -FanMorphN) . x2 implies x1 = x2 ) assume that A3: x1 in dom (cn -FanMorphN) and A4: x2 in dom (cn -FanMorphN) and A5: (cn -FanMorphN) . x1 = (cn -FanMorphN) . x2 ; ::_thesis: x1 = x2 reconsider p2 = x2 as Point of (TOP-REAL 2) by A4; reconsider p1 = x1 as Point of (TOP-REAL 2) by A3; set q = p1; set p = p2; A6: 1 - cn > 0 by A2, XREAL_1:149; now__::_thesis:_(_(_p1_`2_<=_0_&_x1_=_x2_)_or_(_(p1_`1)_/_|.p1.|_>=_cn_&_p1_`2_>=_0_&_p1_<>_0._(TOP-REAL_2)_&_x1_=_x2_)_or_(_(p1_`1)_/_|.p1.|_<_cn_&_p1_`2_>=_0_&_p1_<>_0._(TOP-REAL_2)_&_x1_=_x2_)_) percases ( p1 `2 <= 0 or ( (p1 `1) / |.p1.| >= cn & p1 `2 >= 0 & p1 <> 0. (TOP-REAL 2) ) or ( (p1 `1) / |.p1.| < cn & p1 `2 >= 0 & p1 <> 0. (TOP-REAL 2) ) ) by JGRAPH_2:3; caseA7: p1 `2 <= 0 ; ::_thesis: x1 = x2 then A8: (cn -FanMorphN) . p1 = p1 by Th49; now__::_thesis:_(_(_p2_`2_<=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`1)_/_|.p2.|_>=_cn_&_p2_`2_>=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`1)_/_|.p2.|_<_cn_&_p2_`2_>=_0_&_x1_=_x2_)_) percases ( p2 `2 <= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 >= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 >= 0 ) ) by JGRAPH_2:3; case p2 `2 <= 0 ; ::_thesis: x1 = x2 hence x1 = x2 by A5, A8, Th49; ::_thesis: verum end; caseA9: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 >= 0 ) ; ::_thesis: x1 = x2 set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]|; A10: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; 0 <= (p2 `2) ^2 by XREAL_1:63; then 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) by XREAL_1:7; then A11: ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by A10, XREAL_1:72; A12: |.p2.| > 0 by A9, Lm1; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A11, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p2 `1) / |.p2.| by SQUARE_1:51; then 1 - cn >= ((p2 `1) / |.p2.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((p2 `1) / |.p2.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p2 `1) / |.p2.|) - cn)) / (1 - cn) by A6, XREAL_1:72; then A13: - 1 <= (- (((p2 `1) / |.p2.|) - cn)) / (1 - cn) by A6, XCMPLX_1:197; A14: ((p2 `1) / |.p2.|) - cn >= 0 by A9, XREAL_1:48; A15: (cn -FanMorphN) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| by A1, A2, A9, Th51; ((p2 `1) / |.p2.|) - cn >= 0 by A9, XREAL_1:48; then ((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A6, A13, SQUARE_1:49; then A16: 1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p2 `1) / |.p2.|) - cn)) ^2) / ((1 - cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p2 `1) / |.p2.|) - cn) ^2) / ((1 - cn) ^2))) >= 0 ; then sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)) >= 0 by XCMPLX_1:76; then ( |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| `2 = |.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))) & p1 `2 = 0 ) by A5, A7, A8, A15, EUCLID:52; then A17: sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)) = 0 by A5, A8, A15, A12, XCMPLX_1:6; 1 - ((- ((((p2 `1) / |.p2.|) - cn) / (1 - cn))) ^2) >= 0 by A16, XCMPLX_1:187; then 1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2) = 0 by A17, SQUARE_1:24; then 1 = (((p2 `1) / |.p2.|) - cn) / (1 - cn) by A6, A14, SQUARE_1:18, SQUARE_1:22; then 1 * (1 - cn) = ((p2 `1) / |.p2.|) - cn by A6, XCMPLX_1:87; then 1 * |.p2.| = p2 `1 by A12, XCMPLX_1:87; then p2 `2 = 0 by A10, XCMPLX_1:6; hence x1 = x2 by A5, A8, Th49; ::_thesis: verum end; caseA18: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 >= 0 ) ; ::_thesis: x1 = x2 then A19: |.p2.| <> 0 by TOPRNS_1:24; then A20: |.p2.| ^2 > 0 by SQUARE_1:12; set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]|; A21: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; A22: 1 + cn > 0 by A1, XREAL_1:148; A23: ((p2 `1) / |.p2.|) - cn <= 0 by A18, XREAL_1:47; then A24: - 1 <= (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) by A22; A25: (cn -FanMorphN) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| by A1, A2, A18, Th51; 0 <= (p2 `2) ^2 by XREAL_1:63; then 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) by XREAL_1:7; then ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by A21, XREAL_1:72; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A20, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (- ((p2 `1) / |.p2.|)) ^2 <= 1 ; then 1 >= - ((p2 `1) / |.p2.|) by SQUARE_1:51; then 1 + cn >= (- ((p2 `1) / |.p2.|)) + cn by XREAL_1:7; then (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) <= 1 by A22, XREAL_1:185; then ((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A24, SQUARE_1:49; then A26: 1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p2 `1) / |.p2.|) - cn)) ^2) / ((1 + cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p2 `1) / |.p2.|) - cn) ^2) / ((1 + cn) ^2))) >= 0 ; then sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)) >= 0 by XCMPLX_1:76; then ( |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| `2 = |.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))) & p1 `2 = 0 ) by A5, A7, A8, A25, EUCLID:52; then A27: sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)) = 0 by A5, A8, A25, A19, XCMPLX_1:6; 1 - ((- ((((p2 `1) / |.p2.|) - cn) / (1 + cn))) ^2) >= 0 by A26, XCMPLX_1:187; then 1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2) = 0 by A27, SQUARE_1:24; then 1 = sqrt ((- ((((p2 `1) / |.p2.|) - cn) / (1 + cn))) ^2) by SQUARE_1:18; then 1 = - ((((p2 `1) / |.p2.|) - cn) / (1 + cn)) by A22, A23, SQUARE_1:22; then 1 = (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) by XCMPLX_1:187; then 1 * (1 + cn) = - (((p2 `1) / |.p2.|) - cn) by A22, XCMPLX_1:87; then (1 + cn) - cn = - ((p2 `1) / |.p2.|) ; then 1 = (- (p2 `1)) / |.p2.| by XCMPLX_1:187; then 1 * |.p2.| = - (p2 `1) by A18, TOPRNS_1:24, XCMPLX_1:87; then ((p2 `1) ^2) - ((p2 `1) ^2) = (p2 `2) ^2 by A21, XCMPLX_1:26; then p2 `2 = 0 by XCMPLX_1:6; hence x1 = x2 by A5, A8, Th49; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; caseA28: ( (p1 `1) / |.p1.| >= cn & p1 `2 >= 0 & p1 <> 0. (TOP-REAL 2) ) ; ::_thesis: x1 = x2 then |.p1.| <> 0 by TOPRNS_1:24; then A29: |.p1.| ^2 > 0 by SQUARE_1:12; set q4 = |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]|; A30: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]| `1 = |.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn)) by EUCLID:52; A31: (cn -FanMorphN) . p1 = |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]| by A1, A2, A28, Th51; A32: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]| `2 = |.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))) by EUCLID:52; now__::_thesis:_(_(_p2_`2_<=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`1)_/_|.p2.|_>=_cn_&_p2_`2_>=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`1)_/_|.p2.|_<_cn_&_p2_`2_>=_0_&_x1_=_x2_)_) percases ( p2 `2 <= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 >= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 >= 0 ) ) by JGRAPH_2:3; caseA33: p2 `2 <= 0 ; ::_thesis: x1 = x2 then A34: (cn -FanMorphN) . p2 = p2 by Th49; A35: |.p1.| <> 0 by A28, TOPRNS_1:24; then A36: |.p1.| ^2 > 0 by SQUARE_1:12; A37: ((p1 `1) / |.p1.|) - cn >= 0 by A28, XREAL_1:48; A38: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1; A39: ((p1 `1) / |.p1.|) - cn >= 0 by A28, XREAL_1:48; A40: 1 - cn > 0 by A2, XREAL_1:149; 0 <= (p1 `2) ^2 by XREAL_1:63; then 0 + ((p1 `1) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) by XREAL_1:7; then ((p1 `1) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by A38, XREAL_1:72; then ((p1 `1) ^2) / (|.p1.| ^2) <= 1 by A36, XCMPLX_1:60; then ((p1 `1) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p1 `1) / |.p1.| by SQUARE_1:51; then 1 - cn >= ((p1 `1) / |.p1.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((p1 `1) / |.p1.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p1 `1) / |.p1.|) - cn)) / (1 - cn) by A40, XREAL_1:72; then - 1 <= (- (((p1 `1) / |.p1.|) - cn)) / (1 - cn) by A40, XCMPLX_1:197; then ((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A40, A37, SQUARE_1:49; then A41: 1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p1 `1) / |.p1.|) - cn)) ^2) / ((1 - cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) ^2) / ((1 - cn) ^2))) >= 0 ; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)) >= 0 by XCMPLX_1:76; then p2 `2 = 0 by A5, A31, A33, A34, EUCLID:52; then A42: sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)) = 0 by A5, A31, A32, A34, A35, XCMPLX_1:6; 1 - ((- ((((p1 `1) / |.p1.|) - cn) / (1 - cn))) ^2) >= 0 by A41, XCMPLX_1:187; then 1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2) = 0 by A42, SQUARE_1:24; then 1 = (((p1 `1) / |.p1.|) - cn) / (1 - cn) by A40, A39, SQUARE_1:18, SQUARE_1:22; then 1 * (1 - cn) = ((p1 `1) / |.p1.|) - cn by A40, XCMPLX_1:87; then 1 * |.p1.| = p1 `1 by A28, TOPRNS_1:24, XCMPLX_1:87; then p1 `2 = 0 by A38, XCMPLX_1:6; hence x1 = x2 by A5, A34, Th49; ::_thesis: verum end; caseA43: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 >= 0 ) ; ::_thesis: x1 = x2 0 <= (p1 `2) ^2 by XREAL_1:63; then ( |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) & 0 + ((p1 `1) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p1 `1) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by XREAL_1:72; then ((p1 `1) ^2) / (|.p1.| ^2) <= 1 by A29, XCMPLX_1:60; then ((p1 `1) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p1 `1) / |.p1.| by SQUARE_1:51; then 1 - cn >= ((p1 `1) / |.p1.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((p1 `1) / |.p1.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p1 `1) / |.p1.|) - cn)) / (1 - cn) by A6, XREAL_1:72; then A44: - 1 <= (- (((p1 `1) / |.p1.|) - cn)) / (1 - cn) by A6, XCMPLX_1:197; ((p1 `1) / |.p1.|) - cn >= 0 by A28, XREAL_1:48; then ((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A6, A44, SQUARE_1:49; then 1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A45: 1 - ((- ((((p1 `1) / |.p1.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]| `2 = |.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))) by EUCLID:52; then A46: (|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]| `2) ^2 = (|.p1.| ^2) * ((sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))) ^2) .= (|.p1.| ^2) * (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)) by A45, SQUARE_1:def_2 ; A47: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]| `1 = |.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn)) by EUCLID:52; |.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]|.| ^2 = ((|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]| `1) ^2) + ((|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]| `2) ^2) by JGRAPH_3:1 .= |.p1.| ^2 by A47, A46 ; then A48: sqrt (|.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]|.| ^2) = |.p1.| by SQUARE_1:22; then A49: |.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))))]|.| = |.p1.| by SQUARE_1:22; 0 <= (p2 `2) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A50: ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; |.p2.| <> 0 by A43, TOPRNS_1:24; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A50, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p2 `1) / |.p2.| by SQUARE_1:51; then 1 - cn >= ((p2 `1) / |.p2.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((p2 `1) / |.p2.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p2 `1) / |.p2.|) - cn)) / (1 - cn) by A6, XREAL_1:72; then A51: - 1 <= (- (((p2 `1) / |.p2.|) - cn)) / (1 - cn) by A6, XCMPLX_1:197; ((p2 `1) / |.p2.|) - cn >= 0 by A43, XREAL_1:48; then ((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A6, A51, SQUARE_1:49; then 1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A52: 1 - ((- ((((p2 `1) / |.p2.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]|; A53: |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn)) by EUCLID:52; |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| `2 = |.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))) by EUCLID:52; then A54: (|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| `2) ^2 = (|.p2.| ^2) * ((sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))) ^2) .= (|.p2.| ^2) * (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)) by A52, SQUARE_1:def_2 ; |.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]|.| ^2 = ((|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| `1) ^2) + ((|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| `2) ^2) by JGRAPH_3:1 .= |.p2.| ^2 by A53, A54 ; then A55: sqrt (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]|.| ^2) = |.p2.| by SQUARE_1:22; then A56: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]|.| = |.p2.| by SQUARE_1:22; A57: (cn -FanMorphN) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| by A1, A2, A43, Th51; then (((p2 `1) / |.p2.|) - cn) / (1 - cn) = (|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))) / |.p2.| by A5, A31, A30, A43, A53, TOPRNS_1:24, XCMPLX_1:89; then (((p2 `1) / |.p2.|) - cn) / (1 - cn) = (((p1 `1) / |.p1.|) - cn) / (1 - cn) by A5, A31, A43, A57, A48, A55, TOPRNS_1:24, XCMPLX_1:89; then ((((p2 `1) / |.p2.|) - cn) / (1 - cn)) * (1 - cn) = ((p1 `1) / |.p1.|) - cn by A6, XCMPLX_1:87; then ((p2 `1) / |.p2.|) - cn = ((p1 `1) / |.p1.|) - cn by A6, XCMPLX_1:87; then ((p2 `1) / |.p2.|) * |.p2.| = p1 `1 by A5, A31, A43, A57, A49, A56, TOPRNS_1:24, XCMPLX_1:87; then A58: p2 `1 = p1 `1 by A43, TOPRNS_1:24, XCMPLX_1:87; A59: p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53; ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1; then p2 `2 = sqrt ((p1 `2) ^2) by A5, A31, A43, A57, A49, A56, A58, SQUARE_1:22; then p2 `2 = p1 `2 by A28, SQUARE_1:22; hence x1 = x2 by A58, A59, EUCLID:53; ::_thesis: verum end; caseA60: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 >= 0 ) ; ::_thesis: x1 = x2 then ((p2 `1) / |.p2.|) - cn < 0 by XREAL_1:49; then A61: (((p2 `1) / |.p2.|) - cn) / (1 + cn) < 0 by A1, XREAL_1:141, XREAL_1:148; set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]|; A62: ( |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn)) & ((p1 `1) / |.p1.|) - cn >= 0 ) by A28, EUCLID:52, XREAL_1:48; A63: 1 - cn > 0 by A2, XREAL_1:149; ( (cn -FanMorphN) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| & |.p2.| <> 0 ) by A1, A2, A60, Th51, TOPRNS_1:24; hence x1 = x2 by A5, A31, A30, A61, A62, A63, XREAL_1:132; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; caseA64: ( (p1 `1) / |.p1.| < cn & p1 `2 >= 0 & p1 <> 0. (TOP-REAL 2) ) ; ::_thesis: x1 = x2 then A65: |.p1.| <> 0 by TOPRNS_1:24; then A66: |.p1.| ^2 > 0 by SQUARE_1:12; set q4 = |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]|; A67: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]| `1 = |.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn)) by EUCLID:52; A68: (cn -FanMorphN) . p1 = |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]| by A1, A2, A64, Th51; A69: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]| `2 = |.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))) by EUCLID:52; now__::_thesis:_(_(_p2_`2_<=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`1)_/_|.p2.|_>=_cn_&_p2_`2_>=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`1)_/_|.p2.|_<_cn_&_p2_`2_>=_0_&_x1_=_x2_)_) percases ( p2 `2 <= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 >= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 >= 0 ) ) by JGRAPH_2:3; caseA70: p2 `2 <= 0 ; ::_thesis: x1 = x2 A71: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1; A72: 1 + cn > 0 by A1, XREAL_1:148; 0 <= (p1 `2) ^2 by XREAL_1:63; then 0 + ((p1 `1) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) by XREAL_1:7; then ((p1 `1) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by A71, XREAL_1:72; then ((p1 `1) ^2) / (|.p1.| ^2) <= 1 by A66, XCMPLX_1:60; then ((p1 `1) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then (- ((p1 `1) / |.p1.|)) ^2 <= 1 ; then 1 >= - ((p1 `1) / |.p1.|) by SQUARE_1:51; then 1 + cn >= (- ((p1 `1) / |.p1.|)) + cn by XREAL_1:7; then A73: (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) <= 1 by A72, XREAL_1:185; A74: ((p1 `1) / |.p1.|) - cn <= 0 by A64, XREAL_1:47; then - 1 <= (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) by A72; then ((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A73, SQUARE_1:49; then A75: 1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then A76: 1 - ((- ((((p1 `1) / |.p1.|) - cn) / (1 + cn))) ^2) >= 0 by XCMPLX_1:187; A77: (cn -FanMorphN) . p2 = p2 by A70, Th49; sqrt (1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2)) >= 0 by A75, SQUARE_1:def_2; then sqrt (1 - (((- (((p1 `1) / |.p1.|) - cn)) ^2) / ((1 + cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) ^2) / ((1 + cn) ^2))) >= 0 ; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)) >= 0 by XCMPLX_1:76; then p2 `2 = 0 by A5, A68, A70, A77, EUCLID:52; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)) = 0 by A5, A68, A69, A65, A77, XCMPLX_1:6; then 1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2) = 0 by A76, SQUARE_1:24; then 1 = sqrt ((- ((((p1 `1) / |.p1.|) - cn) / (1 + cn))) ^2) by SQUARE_1:18; then 1 = - ((((p1 `1) / |.p1.|) - cn) / (1 + cn)) by A72, A74, SQUARE_1:22; then 1 = (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) by XCMPLX_1:187; then 1 * (1 + cn) = - (((p1 `1) / |.p1.|) - cn) by A72, XCMPLX_1:87; then (1 + cn) - cn = - ((p1 `1) / |.p1.|) ; then 1 = (- (p1 `1)) / |.p1.| by XCMPLX_1:187; then 1 * |.p1.| = - (p1 `1) by A64, TOPRNS_1:24, XCMPLX_1:87; then ((p1 `1) ^2) - ((p1 `1) ^2) = (p1 `2) ^2 by A71, XCMPLX_1:26; then p1 `2 = 0 by XCMPLX_1:6; hence x1 = x2 by A5, A77, Th49; ::_thesis: verum end; caseA78: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 >= 0 ) ; ::_thesis: x1 = x2 set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]|; A79: ( |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn)) & |.p1.| <> 0 ) by A64, EUCLID:52, TOPRNS_1:24; ((p1 `1) / |.p1.|) - cn < 0 by A64, XREAL_1:49; then A80: (((p1 `1) / |.p1.|) - cn) / (1 + cn) < 0 by A1, XREAL_1:141, XREAL_1:148; A81: 1 - cn > 0 by A2, XREAL_1:149; ( (cn -FanMorphN) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))))]| & ((p2 `1) / |.p2.|) - cn >= 0 ) by A1, A2, A78, Th51, XREAL_1:48; hence x1 = x2 by A5, A68, A67, A80, A79, A81, XREAL_1:132; ::_thesis: verum end; caseA82: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 >= 0 ) ; ::_thesis: x1 = x2 0 <= (p2 `2) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A83: ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; A84: 1 + cn > 0 by A1, XREAL_1:148; 0 <= (p1 `2) ^2 by XREAL_1:63; then ( |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) & 0 + ((p1 `1) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p1 `1) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by XREAL_1:72; then ((p1 `1) ^2) / (|.p1.| ^2) <= 1 by A66, XCMPLX_1:60; then ((p1 `1) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p1 `1) / |.p1.| by SQUARE_1:51; then (- 1) - cn <= ((p1 `1) / |.p1.|) - cn by XREAL_1:9; then - ((- 1) - cn) >= - (((p1 `1) / |.p1.|) - cn) by XREAL_1:24; then A85: (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) <= 1 by A84, XREAL_1:185; ((p1 `1) / |.p1.|) - cn <= 0 by A64, XREAL_1:47; then - 1 <= (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) by A84; then ((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A85, SQUARE_1:49; then 1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then A86: 1 - ((- ((((p1 `1) / |.p1.|) - cn) / (1 + cn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]| `2 = |.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))) by EUCLID:52; then A87: (|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]| `2) ^2 = (|.p1.| ^2) * ((sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))) ^2) .= (|.p1.| ^2) * (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)) by A86, SQUARE_1:def_2 ; A88: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]| `1 = |.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn)) by EUCLID:52; set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]|; A89: |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn)) by EUCLID:52; |.p2.| <> 0 by A82, TOPRNS_1:24; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A83, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p2 `1) / |.p2.| by SQUARE_1:51; then (- 1) - cn <= ((p2 `1) / |.p2.|) - cn by XREAL_1:9; then - ((- 1) - cn) >= - (((p2 `1) / |.p2.|) - cn) by XREAL_1:24; then A90: (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) <= 1 by A84, XREAL_1:185; ((p2 `1) / |.p2.|) - cn <= 0 by A82, XREAL_1:47; then - 1 <= (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) by A84; then ((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A90, SQUARE_1:49; then 1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then A91: 1 - ((- ((((p2 `1) / |.p2.|) - cn) / (1 + cn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| `2 = |.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))) by EUCLID:52; then A92: (|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| `2) ^2 = (|.p2.| ^2) * ((sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))) ^2) .= (|.p2.| ^2) * (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)) by A91, SQUARE_1:def_2 ; |.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]|.| ^2 = ((|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| `1) ^2) + ((|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| `2) ^2) by JGRAPH_3:1 .= |.p2.| ^2 by A89, A92 ; then A93: sqrt (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]|.| ^2) = |.p2.| by SQUARE_1:22; then A94: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]|.| = |.p2.| by SQUARE_1:22; |.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]|.| ^2 = ((|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]| `1) ^2) + ((|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]| `2) ^2) by JGRAPH_3:1 .= |.p1.| ^2 by A88, A87 ; then A95: sqrt (|.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]|.| ^2) = |.p1.| by SQUARE_1:22; then A96: |.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))))]|.| = |.p1.| by SQUARE_1:22; A97: (cn -FanMorphN) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))))]| by A1, A2, A82, Th51; then (((p2 `1) / |.p2.|) - cn) / (1 + cn) = (|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))) / |.p2.| by A5, A68, A67, A82, A89, TOPRNS_1:24, XCMPLX_1:89; then (((p2 `1) / |.p2.|) - cn) / (1 + cn) = (((p1 `1) / |.p1.|) - cn) / (1 + cn) by A5, A68, A82, A97, A95, A93, TOPRNS_1:24, XCMPLX_1:89; then ((((p2 `1) / |.p2.|) - cn) / (1 + cn)) * (1 + cn) = ((p1 `1) / |.p1.|) - cn by A84, XCMPLX_1:87; then ((p2 `1) / |.p2.|) - cn = ((p1 `1) / |.p1.|) - cn by A84, XCMPLX_1:87; then ((p2 `1) / |.p2.|) * |.p2.| = p1 `1 by A5, A68, A82, A97, A96, A94, TOPRNS_1:24, XCMPLX_1:87; then A98: p2 `1 = p1 `1 by A82, TOPRNS_1:24, XCMPLX_1:87; A99: p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53; ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1; then p2 `2 = sqrt ((p1 `2) ^2) by A5, A68, A82, A97, A96, A94, A98, SQUARE_1:22; then p2 `2 = p1 `2 by A64, SQUARE_1:22; hence x1 = x2 by A98, A99, EUCLID:53; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; hence cn -FanMorphN is one-to-one by FUNCT_1:def_4; ::_thesis: verum end; theorem Th72: :: JGRAPH_4:72 for cn being Real st - 1 < cn & cn < 1 holds ( cn -FanMorphN is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) ) proof let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies ( cn -FanMorphN is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) ) ) assume that A1: - 1 < cn and A2: cn < 1 ; ::_thesis: ( cn -FanMorphN is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) ) thus cn -FanMorphN is Function of (TOP-REAL 2),(TOP-REAL 2) ; ::_thesis: rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = cn -FanMorphN holds rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) proof let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = cn -FanMorphN implies rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) ) assume A3: f = cn -FanMorphN ; ::_thesis: rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) A4: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; the carrier of (TOP-REAL 2) c= rng f proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (TOP-REAL 2) or y in rng f ) assume y in the carrier of (TOP-REAL 2) ; ::_thesis: y in rng f then reconsider p2 = y as Point of (TOP-REAL 2) ; set q = p2; now__::_thesis:_(_(_p2_`2_<=_0_&_ex_x_being_set_st_ (_x_in_dom_(cn_-FanMorphN)_&_y_=_(cn_-FanMorphN)_._x_)_)_or_(_(p2_`1)_/_|.p2.|_>=_0_&_p2_`2_>=_0_&_p2_<>_0._(TOP-REAL_2)_&_ex_x_being_set_st_ (_x_in_dom_(cn_-FanMorphN)_&_y_=_(cn_-FanMorphN)_._x_)_)_or_(_(p2_`1)_/_|.p2.|_<_0_&_p2_`2_>=_0_&_p2_<>_0._(TOP-REAL_2)_&_ex_x_being_set_st_ (_x_in_dom_(cn_-FanMorphN)_&_y_=_(cn_-FanMorphN)_._x_)_)_) percases ( p2 `2 <= 0 or ( (p2 `1) / |.p2.| >= 0 & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) or ( (p2 `1) / |.p2.| < 0 & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) ) by JGRAPH_2:3; case p2 `2 <= 0 ; ::_thesis: ex x being set st ( x in dom (cn -FanMorphN) & y = (cn -FanMorphN) . x ) then y = (cn -FanMorphN) . p2 by Th49; hence ex x being set st ( x in dom (cn -FanMorphN) & y = (cn -FanMorphN) . x ) by A3, A4; ::_thesis: verum end; caseA5: ( (p2 `1) / |.p2.| >= 0 & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st ( x in dom (cn -FanMorphN) & y = (cn -FanMorphN) . x ) - (- (1 + cn)) > 0 by A1, XREAL_1:148; then A6: - ((- 1) - cn) > 0 ; A7: 1 - cn >= 0 by A2, XREAL_1:149; then ((p2 `1) / |.p2.|) * (1 - cn) >= 0 by A5; then (- 1) - cn <= ((p2 `1) / |.p2.|) * (1 - cn) by A6; then A8: ((- 1) - cn) + cn <= (((p2 `1) / |.p2.|) * (1 - cn)) + cn by XREAL_1:7; set px = |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|; A9: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn) by EUCLID:52; |.p2.| <> 0 by A5, TOPRNS_1:24; then A10: |.p2.| ^2 > 0 by SQUARE_1:12; A11: dom (cn -FanMorphN) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A12: 1 - cn > 0 by A2, XREAL_1:149; 0 <= (p2 `2) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A10, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (p2 `1) / |.p2.| <= 1 by SQUARE_1:51; then ((p2 `1) / |.p2.|) * (1 - cn) <= 1 * (1 - cn) by A12, XREAL_1:64; then ((((p2 `1) / |.p2.|) * (1 - cn)) + cn) - cn <= 1 - cn ; then (((p2 `1) / |.p2.|) * (1 - cn)) + cn <= 1 by XREAL_1:9; then 1 ^2 >= ((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2 by A8, SQUARE_1:49; then A13: 1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2) >= 0 by XREAL_1:48; then A14: sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)) >= 0 by SQUARE_1:def_2; A15: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| `2 = |.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))) by EUCLID:52; then |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2 = ((|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))) ^2) + ((|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)) ^2) by A9, JGRAPH_3:1 .= ((|.p2.| ^2) * ((sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))) ^2)) + ((|.p2.| ^2) * (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)) ; then A16: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2 = ((|.p2.| ^2) * (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))) + ((|.p2.| ^2) * (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)) by A13, SQUARE_1:def_2 .= |.p2.| ^2 ; then A17: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| = sqrt (|.p2.| ^2) by SQUARE_1:22 .= |.p2.| by SQUARE_1:22 ; then A18: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| <> 0. (TOP-REAL 2) by A5, TOPRNS_1:23, TOPRNS_1:24; (((p2 `1) / |.p2.|) * (1 - cn)) + cn >= 0 + cn by A5, A7, XREAL_1:7; then (|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| >= cn by A5, A9, A17, TOPRNS_1:24, XCMPLX_1:89; then A19: (cn -FanMorphN) . |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| = |[(|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * ((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.|) - cn) / (1 - cn))),(|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (1 - (((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.|) - cn) / (1 - cn)) ^2))))]| by A1, A2, A15, A14, A18, Th51; A20: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (((p2 `2) / |.p2.|) ^2)) = |.p2.| * ((p2 `2) / |.p2.|) by A5, A17, SQUARE_1:22 .= p2 `2 by A5, TOPRNS_1:24, XCMPLX_1:87 ; A21: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * ((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.|) - cn) / (1 - cn)) = |.p2.| * ((((((p2 `1) / |.p2.|) * (1 - cn)) + cn) - cn) / (1 - cn)) by A5, A9, A17, TOPRNS_1:24, XCMPLX_1:89 .= |.p2.| * ((p2 `1) / |.p2.|) by A12, XCMPLX_1:89 .= p2 `1 by A5, TOPRNS_1:24, XCMPLX_1:87 ; then |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (1 - (((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.|) - cn) / (1 - cn)) ^2))) = |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (1 - (((p2 `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.|) ^2))) by A5, A17, TOPRNS_1:24, XCMPLX_1:89 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (1 - (((p2 `1) ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2)))) by XCMPLX_1:76 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (((|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2)) - (((p2 `1) ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2)))) by A10, A16, XCMPLX_1:60 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (((|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2) - ((p2 `1) ^2)) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2))) by XCMPLX_1:120 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (((((p2 `1) ^2) + ((p2 `2) ^2)) - ((p2 `1) ^2)) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| ^2))) by A16, JGRAPH_3:1 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))))]|.| * (sqrt (((p2 `2) / |.p2.|) ^2)) by A17, XCMPLX_1:76 ; hence ex x being set st ( x in dom (cn -FanMorphN) & y = (cn -FanMorphN) . x ) by A19, A21, A20, A11, EUCLID:53; ::_thesis: verum end; caseA22: ( (p2 `1) / |.p2.| < 0 & p2 `2 >= 0 & p2 <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st ( x in dom (cn -FanMorphN) & y = (cn -FanMorphN) . x ) A23: 1 + cn >= 0 by A1, XREAL_1:148; 1 - cn > 0 by A2, XREAL_1:149; then A24: (1 - cn) + cn >= (((p2 `1) / |.p2.|) * (1 + cn)) + cn by A22, A23, XREAL_1:7; A25: 1 + cn > 0 by A1, XREAL_1:148; |.p2.| <> 0 by A22, TOPRNS_1:24; then A26: |.p2.| ^2 > 0 by SQUARE_1:12; 0 <= (p2 `2) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A26, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (p2 `1) / |.p2.| >= - 1 by SQUARE_1:51; then ((p2 `1) / |.p2.|) * (1 + cn) >= (- 1) * (1 + cn) by A25, XREAL_1:64; then ((((p2 `1) / |.p2.|) * (1 + cn)) + cn) - cn >= (- 1) - cn ; then (((p2 `1) / |.p2.|) * (1 + cn)) + cn >= - 1 by XREAL_1:9; then 1 ^2 >= ((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2 by A24, SQUARE_1:49; then A27: 1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2) >= 0 by XREAL_1:48; then A28: sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)) >= 0 by SQUARE_1:def_2; A29: dom (cn -FanMorphN) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; set px = |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|; A30: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn) by EUCLID:52; A31: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| `2 = |.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))) by EUCLID:52; then |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2 = ((|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))) ^2) + ((|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)) ^2) by A30, JGRAPH_3:1 .= ((|.p2.| ^2) * ((sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))) ^2)) + ((|.p2.| ^2) * (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)) ; then A32: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2 = ((|.p2.| ^2) * (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))) + ((|.p2.| ^2) * (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)) by A27, SQUARE_1:def_2 .= |.p2.| ^2 ; then A33: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| = sqrt (|.p2.| ^2) by SQUARE_1:22 .= |.p2.| by SQUARE_1:22 ; then A34: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| <> 0. (TOP-REAL 2) by A22, TOPRNS_1:23, TOPRNS_1:24; (((p2 `1) / |.p2.|) * (1 + cn)) + cn <= 0 + cn by A22, A23, XREAL_1:7; then (|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| <= cn by A22, A30, A33, TOPRNS_1:24, XCMPLX_1:89; then A35: (cn -FanMorphN) . |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| = |[(|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * ((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.|) - cn) / (1 + cn))),(|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (1 - (((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.|) - cn) / (1 + cn)) ^2))))]| by A1, A2, A31, A28, A34, Th51; A36: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (((p2 `2) / |.p2.|) ^2)) = |.p2.| * ((p2 `2) / |.p2.|) by A22, A33, SQUARE_1:22 .= p2 `2 by A22, TOPRNS_1:24, XCMPLX_1:87 ; A37: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * ((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.|) - cn) / (1 + cn)) = |.p2.| * ((((((p2 `1) / |.p2.|) * (1 + cn)) + cn) - cn) / (1 + cn)) by A22, A30, A33, TOPRNS_1:24, XCMPLX_1:89 .= |.p2.| * ((p2 `1) / |.p2.|) by A25, XCMPLX_1:89 .= p2 `1 by A22, TOPRNS_1:24, XCMPLX_1:87 ; then |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (1 - (((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.|) - cn) / (1 + cn)) ^2))) = |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (1 - (((p2 `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.|) ^2))) by A22, A33, TOPRNS_1:24, XCMPLX_1:89 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (1 - (((p2 `1) ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2)))) by XCMPLX_1:76 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (((|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2)) - (((p2 `1) ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2)))) by A26, A32, XCMPLX_1:60 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (((|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2) - ((p2 `1) ^2)) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2))) by XCMPLX_1:120 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (((((p2 `1) ^2) + ((p2 `2) ^2)) - ((p2 `1) ^2)) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| ^2))) by A32, JGRAPH_3:1 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))]|.| * (sqrt (((p2 `2) / |.p2.|) ^2)) by A33, XCMPLX_1:76 ; hence ex x being set st ( x in dom (cn -FanMorphN) & y = (cn -FanMorphN) . x ) by A35, A37, A36, A29, EUCLID:53; ::_thesis: verum end; end; end; hence y in rng f by A3, FUNCT_1:def_3; ::_thesis: verum end; hence rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) by A3, XBOOLE_0:def_10; ::_thesis: verum end; hence rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) ; ::_thesis: verum end; theorem Th73: :: JGRAPH_4:73 for cn being Real for p2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 holds ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphN) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphN) . p2 in V2 ) ) proof reconsider O = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:67; let cn be Real; ::_thesis: for p2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 holds ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphN) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphN) . p2 in V2 ) ) let p2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 implies ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphN) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphN) . p2 in V2 ) ) ) A1: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider V0 = Ball (O,(|.p2.| + 1)) as Subset of (TOP-REAL 2) ; ( O in V0 & V0 c= cl_Ball (O,(|.p2.| + 1)) ) by GOBOARD6:1, METRIC_1:14; then reconsider K0 = cl_Ball (O,(|.p2.| + 1)) as non empty compact Subset of (TOP-REAL 2) by A1, Th15; set q3 = (cn -FanMorphN) . p2; reconsider VV0 = V0 as Subset of (TopSpaceMetr (Euclid 2)) ; reconsider u2 = p2 as Point of (Euclid 2) by EUCLID:67; reconsider u3 = (cn -FanMorphN) . p2 as Point of (Euclid 2) by EUCLID:67; A2: (cn -FanMorphN) .: K0 c= K0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (cn -FanMorphN) .: K0 or y in K0 ) assume y in (cn -FanMorphN) .: K0 ; ::_thesis: y in K0 then consider x being set such that A3: x in dom (cn -FanMorphN) and A4: x in K0 and A5: y = (cn -FanMorphN) . x by FUNCT_1:def_6; reconsider q = x as Point of (TOP-REAL 2) by A3; reconsider uq = q as Point of (Euclid 2) by EUCLID:67; dist (O,uq) <= |.p2.| + 1 by A4, METRIC_1:12; then |.((0. (TOP-REAL 2)) - q).| <= |.p2.| + 1 by JGRAPH_1:28; then |.(- q).| <= |.p2.| + 1 by EUCLID:27; then A6: |.q.| <= |.p2.| + 1 by TOPRNS_1:26; A7: y in rng (cn -FanMorphN) by A3, A5, FUNCT_1:def_3; then reconsider u = y as Point of (Euclid 2) by EUCLID:67; reconsider q4 = y as Point of (TOP-REAL 2) by A7; |.q4.| = |.q.| by A5, Th66; then |.(- q4).| <= |.p2.| + 1 by A6, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - q4).| <= |.p2.| + 1 by EUCLID:27; then dist (O,u) <= |.p2.| + 1 by JGRAPH_1:28; hence y in K0 by METRIC_1:12; ::_thesis: verum end; VV0 is open by TOPMETR:14; then A8: V0 is open by Lm11, PRE_TOPC:30; A9: |.p2.| < |.p2.| + 1 by XREAL_1:29; then |.(- p2).| < |.p2.| + 1 by TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - p2).| < |.p2.| + 1 by EUCLID:27; then dist (O,u2) < |.p2.| + 1 by JGRAPH_1:28; then A10: p2 in V0 by METRIC_1:11; |.((cn -FanMorphN) . p2).| = |.p2.| by Th66; then |.(- ((cn -FanMorphN) . p2)).| < |.p2.| + 1 by A9, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - ((cn -FanMorphN) . p2)).| < |.p2.| + 1 by EUCLID:27; then dist (O,u3) < |.p2.| + 1 by JGRAPH_1:28; then A11: (cn -FanMorphN) . p2 in V0 by METRIC_1:11; assume A12: ( - 1 < cn & cn < 1 ) ; ::_thesis: ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphN) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphN) . p2 in V2 ) ) K0 c= (cn -FanMorphN) .: K0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in K0 or y in (cn -FanMorphN) .: K0 ) assume A13: y in K0 ; ::_thesis: y in (cn -FanMorphN) .: K0 then reconsider q4 = y as Point of (TOP-REAL 2) ; reconsider y = y as Point of (Euclid 2) by A13; the carrier of (TOP-REAL 2) c= rng (cn -FanMorphN) by A12, Th72; then q4 in rng (cn -FanMorphN) by TARSKI:def_3; then consider x being set such that A14: x in dom (cn -FanMorphN) and A15: y = (cn -FanMorphN) . x by FUNCT_1:def_3; reconsider x = x as Point of (Euclid 2) by A14, Lm11; reconsider q = x as Point of (TOP-REAL 2) by A14; |.q4.| = |.q.| by A15, Th66; then q in K0 by A13, Lm12; hence y in (cn -FanMorphN) .: K0 by A14, A15, FUNCT_1:def_6; ::_thesis: verum end; then K0 = (cn -FanMorphN) .: K0 by A2, XBOOLE_0:def_10; hence ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphN) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphN) . p2 in V2 ) ) by A10, A8, A11, METRIC_1:14; ::_thesis: verum end; theorem :: JGRAPH_4:74 for cn being Real st - 1 < cn & cn < 1 holds ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = cn -FanMorphN & f is being_homeomorphism ) proof let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = cn -FanMorphN & f is being_homeomorphism ) ) reconsider f = cn -FanMorphN as Function of (TOP-REAL 2),(TOP-REAL 2) ; assume A1: ( - 1 < cn & cn < 1 ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = cn -FanMorphN & f is being_homeomorphism ) then A2: for p2 being Point of (TOP-REAL 2) ex K being non empty compact Subset of (TOP-REAL 2) st ( K = f .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & f . p2 in V2 ) ) by Th73; ( rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) & ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = cn -FanMorphN & h is continuous ) ) by A1, Th70, Th72; then f is being_homeomorphism by A1, A2, Th3, Th71; hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = cn -FanMorphN & f is being_homeomorphism ) ; ::_thesis: verum end; theorem Th75: :: JGRAPH_4:75 for cn being Real for q being Point of (TOP-REAL 2) st cn < 1 & q `2 > 0 & (q `1) / |.q.| >= cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 >= 0 ) proof let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st cn < 1 & q `2 > 0 & (q `1) / |.q.| >= cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 >= 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( cn < 1 & q `2 > 0 & (q `1) / |.q.| >= cn implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 >= 0 ) ) assume that A1: cn < 1 and A2: q `2 > 0 and A3: (q `1) / |.q.| >= cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 >= 0 ) A4: ((q `1) / |.q.|) - cn >= 0 by A3, XREAL_1:48; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphN) . q implies ( p `2 > 0 & p `1 >= 0 ) ) set qz = p; A5: 1 - cn > 0 by A1, XREAL_1:149; A6: |.q.| <> 0 by A2, JGRAPH_2:3, TOPRNS_1:24; then A7: |.q.| ^2 > 0 by SQUARE_1:12; ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) < ((q `1) ^2) + ((q `2) ^2) ) by A2, JGRAPH_3:1, SQUARE_1:12, XREAL_1:8; then ((q `1) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A7, XREAL_1:74; then ((q `1) ^2) / (|.q.| ^2) < 1 by A7, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 < 1 by XCMPLX_1:76; then 1 > (q `1) / |.q.| by SQUARE_1:52; then 1 - cn > ((q `1) / |.q.|) - cn by XREAL_1:9; then - (1 - cn) < - (((q `1) / |.q.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) < (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A5, XREAL_1:74; then - 1 < (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A5, XCMPLX_1:197; then ((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2 < 1 ^2 by A5, A4, SQUARE_1:50; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2)) > 0 by SQUARE_1:25; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) ^2) / ((1 - cn) ^2))) > 0 by XCMPLX_1:76; then sqrt (1 - (((((q `1) / |.q.|) - cn) ^2) / ((1 - cn) ^2))) > 0 ; then A8: sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) > 0 by XCMPLX_1:76; assume p = (cn -FanMorphN) . q ; ::_thesis: ( p `2 > 0 & p `1 >= 0 ) then A9: p = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by A2, A3, Th49; then p `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) by EUCLID:52; hence ( p `2 > 0 & p `1 >= 0 ) by A9, A6, A5, A4, A8, EUCLID:52, XREAL_1:129; ::_thesis: verum end; theorem Th76: :: JGRAPH_4:76 for cn being Real for q being Point of (TOP-REAL 2) st - 1 < cn & q `2 > 0 & (q `1) / |.q.| < cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 < 0 ) proof let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < cn & q `2 > 0 & (q `1) / |.q.| < cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 < 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & q `2 > 0 & (q `1) / |.q.| < cn implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 < 0 ) ) assume that A1: - 1 < cn and A2: q `2 > 0 and A3: (q `1) / |.q.| < cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 < 0 ) A4: 1 + cn > 0 by A1, XREAL_1:148; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphN) . q implies ( p `2 > 0 & p `1 < 0 ) ) set qz = p; assume p = (cn -FanMorphN) . q ; ::_thesis: ( p `2 > 0 & p `1 < 0 ) then p = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| by A2, A3, Th50; then A5: ( p `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) & p `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) ) by EUCLID:52; A6: |.q.| <> 0 by A2, JGRAPH_2:3, TOPRNS_1:24; then A7: |.q.| ^2 > 0 by SQUARE_1:12; A8: ((q `1) / |.q.|) - cn < 0 by A3, XREAL_1:49; then - (((q `1) / |.q.|) - cn) > 0 by XREAL_1:58; then (- (1 + cn)) / (1 + cn) < (- (((q `1) / |.q.|) - cn)) / (1 + cn) by A4, XREAL_1:74; then A9: - 1 < (- (((q `1) / |.q.|) - cn)) / (1 + cn) by A4, XCMPLX_1:197; ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) < ((q `1) ^2) + ((q `2) ^2) ) by A2, JGRAPH_3:1, SQUARE_1:12, XREAL_1:8; then ((q `1) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A7, XREAL_1:74; then ((q `1) ^2) / (|.q.| ^2) < 1 by A7, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (q `1) / |.q.| by SQUARE_1:52; then (- 1) - cn < ((q `1) / |.q.|) - cn by XREAL_1:9; then - (- (1 + cn)) > - (((q `1) / |.q.|) - cn) by XREAL_1:24; then (- (((q `1) / |.q.|) - cn)) / (1 + cn) < 1 by A4, XREAL_1:191; then ((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2 < 1 ^2 by A9, SQUARE_1:50; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2)) > 0 by SQUARE_1:25; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) ^2) / ((1 + cn) ^2))) > 0 by XCMPLX_1:76; then sqrt (1 - (((((q `1) / |.q.|) - cn) ^2) / ((1 + cn) ^2))) > 0 ; then A10: sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) > 0 by XCMPLX_1:76; (((q `1) / |.q.|) - cn) / (1 + cn) < 0 by A1, A8, XREAL_1:141, XREAL_1:148; hence ( p `2 > 0 & p `1 < 0 ) by A6, A5, A10, XREAL_1:129, XREAL_1:132; ::_thesis: verum end; theorem Th77: :: JGRAPH_4:77 for cn being Real for q1, q2 being Point of (TOP-REAL 2) st cn < 1 & q1 `2 > 0 & (q1 `1) / |.q1.| >= cn & q2 `2 > 0 & (q2 `1) / |.q2.| >= cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| proof let cn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st cn < 1 & q1 `2 > 0 & (q1 `1) / |.q1.| >= cn & q2 `2 > 0 & (q2 `1) / |.q2.| >= cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( cn < 1 & q1 `2 > 0 & (q1 `1) / |.q1.| >= cn & q2 `2 > 0 & (q2 `1) / |.q2.| >= cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A1: cn < 1 and A2: q1 `2 > 0 and A3: (q1 `1) / |.q1.| >= cn and A4: q2 `2 > 0 and A5: (q2 `1) / |.q2.| >= cn and A6: (q1 `1) / |.q1.| < (q2 `1) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| A7: ( ((q1 `1) / |.q1.|) - cn < ((q2 `1) / |.q2.|) - cn & 1 - cn > 0 ) by A1, A6, XREAL_1:9, XREAL_1:149; let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A8: p1 = (cn -FanMorphN) . q1 and A9: p2 = (cn -FanMorphN) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| A10: |.p2.| = |.q2.| by A9, Th66; p2 = |[(|.q2.| * ((((q2 `1) / |.q2.|) - cn) / (1 - cn))),(|.q2.| * (sqrt (1 - (((((q2 `1) / |.q2.|) - cn) / (1 - cn)) ^2))))]| by A4, A5, A9, Th49; then A11: p2 `1 = |.q2.| * ((((q2 `1) / |.q2.|) - cn) / (1 - cn)) by EUCLID:52; |.q2.| > 0 by A4, Lm1, JGRAPH_2:3; then A12: (p2 `1) / |.p2.| = (((q2 `1) / |.q2.|) - cn) / (1 - cn) by A11, A10, XCMPLX_1:89; p1 = |[(|.q1.| * ((((q1 `1) / |.q1.|) - cn) / (1 - cn))),(|.q1.| * (sqrt (1 - (((((q1 `1) / |.q1.|) - cn) / (1 - cn)) ^2))))]| by A2, A3, A8, Th49; then A13: p1 `1 = |.q1.| * ((((q1 `1) / |.q1.|) - cn) / (1 - cn)) by EUCLID:52; A14: |.p1.| = |.q1.| by A8, Th66; |.q1.| > 0 by A2, Lm1, JGRAPH_2:3; then (p1 `1) / |.p1.| = (((q1 `1) / |.q1.|) - cn) / (1 - cn) by A13, A14, XCMPLX_1:89; hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A12, A7, XREAL_1:74; ::_thesis: verum end; theorem Th78: :: JGRAPH_4:78 for cn being Real for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & q1 `2 > 0 & (q1 `1) / |.q1.| < cn & q2 `2 > 0 & (q2 `1) / |.q2.| < cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| proof let cn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & q1 `2 > 0 & (q1 `1) / |.q1.| < cn & q2 `2 > 0 & (q2 `1) / |.q2.| < cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & q1 `2 > 0 & (q1 `1) / |.q1.| < cn & q2 `2 > 0 & (q2 `1) / |.q2.| < cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A1: - 1 < cn and A2: q1 `2 > 0 and A3: (q1 `1) / |.q1.| < cn and A4: q2 `2 > 0 and A5: (q2 `1) / |.q2.| < cn and A6: (q1 `1) / |.q1.| < (q2 `1) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| A7: ( ((q1 `1) / |.q1.|) - cn < ((q2 `1) / |.q2.|) - cn & 1 + cn > 0 ) by A1, A6, XREAL_1:9, XREAL_1:148; let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A8: p1 = (cn -FanMorphN) . q1 and A9: p2 = (cn -FanMorphN) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| A10: |.p2.| = |.q2.| by A9, Th66; p2 = |[(|.q2.| * ((((q2 `1) / |.q2.|) - cn) / (1 + cn))),(|.q2.| * (sqrt (1 - (((((q2 `1) / |.q2.|) - cn) / (1 + cn)) ^2))))]| by A4, A5, A9, Th50; then A11: p2 `1 = |.q2.| * ((((q2 `1) / |.q2.|) - cn) / (1 + cn)) by EUCLID:52; |.q2.| > 0 by A4, Lm1, JGRAPH_2:3; then A12: (p2 `1) / |.p2.| = (((q2 `1) / |.q2.|) - cn) / (1 + cn) by A11, A10, XCMPLX_1:89; p1 = |[(|.q1.| * ((((q1 `1) / |.q1.|) - cn) / (1 + cn))),(|.q1.| * (sqrt (1 - (((((q1 `1) / |.q1.|) - cn) / (1 + cn)) ^2))))]| by A2, A3, A8, Th50; then A13: p1 `1 = |.q1.| * ((((q1 `1) / |.q1.|) - cn) / (1 + cn)) by EUCLID:52; A14: |.p1.| = |.q1.| by A8, Th66; |.q1.| > 0 by A2, Lm1, JGRAPH_2:3; then (p1 `1) / |.p1.| = (((q1 `1) / |.q1.|) - cn) / (1 + cn) by A13, A14, XCMPLX_1:89; hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A12, A7, XREAL_1:74; ::_thesis: verum end; theorem :: JGRAPH_4:79 for cn being Real for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 > 0 & q2 `2 > 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| proof let cn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 > 0 & q2 `2 > 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q1 `2 > 0 & q2 `2 > 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A1: - 1 < cn and A2: cn < 1 and A3: q1 `2 > 0 and A4: q2 `2 > 0 and A5: (q1 `1) / |.q1.| < (q2 `1) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A6: p1 = (cn -FanMorphN) . q1 and A7: p2 = (cn -FanMorphN) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| percases ( ( (q1 `1) / |.q1.| >= cn & (q2 `1) / |.q2.| >= cn ) or ( (q1 `1) / |.q1.| >= cn & (q2 `1) / |.q2.| < cn ) or ( (q1 `1) / |.q1.| < cn & (q2 `1) / |.q2.| >= cn ) or ( (q1 `1) / |.q1.| < cn & (q2 `1) / |.q2.| < cn ) ) ; suppose ( (q1 `1) / |.q1.| >= cn & (q2 `1) / |.q2.| >= cn ) ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A2, A3, A4, A5, A6, A7, Th77; ::_thesis: verum end; suppose ( (q1 `1) / |.q1.| >= cn & (q2 `1) / |.q2.| < cn ) ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A5, XXREAL_0:2; ::_thesis: verum end; supposeA8: ( (q1 `1) / |.q1.| < cn & (q2 `1) / |.q2.| >= cn ) ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| then p2 `1 >= 0 by A2, A4, A7, Th75; then A9: (p2 `1) / |.p2.| >= 0 ; p1 `1 < 0 by A1, A3, A6, A8, Th76; hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A9, Lm1, JGRAPH_2:3, XREAL_1:141; ::_thesis: verum end; suppose ( (q1 `1) / |.q1.| < cn & (q2 `1) / |.q2.| < cn ) ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A1, A3, A4, A5, A6, A7, Th78; ::_thesis: verum end; end; end; theorem :: JGRAPH_4:80 for cn being Real for q being Point of (TOP-REAL 2) st q `2 > 0 & (q `1) / |.q.| = cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 = 0 ) proof let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st q `2 > 0 & (q `1) / |.q.| = cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 = 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( q `2 > 0 & (q `1) / |.q.| = cn implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 = 0 ) ) assume that A1: q `2 > 0 and A2: (q `1) / |.q.| = cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds ( p `2 > 0 & p `1 = 0 ) A3: ( |.q.| <> 0 & sqrt (1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2)) > 0 ) by A1, A2, JGRAPH_2:3, SQUARE_1:25, TOPRNS_1:24; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphN) . q implies ( p `2 > 0 & p `1 = 0 ) ) assume p = (cn -FanMorphN) . q ; ::_thesis: ( p `2 > 0 & p `1 = 0 ) then A4: p = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by A1, A2, Th49; then p `2 = |.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) by EUCLID:52; hence ( p `2 > 0 & p `1 = 0 ) by A2, A4, A3, EUCLID:52, XREAL_1:129; ::_thesis: verum end; theorem :: JGRAPH_4:81 for cn being real number holds 0. (TOP-REAL 2) = (cn -FanMorphN) . (0. (TOP-REAL 2)) by Th49, JGRAPH_2:3; begin definition let s be real number ; let q be Point of (TOP-REAL 2); func FanE (s,q) -> Point of (TOP-REAL 2) equals :Def6: :: JGRAPH_4:def 6 |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2))),((((q `2) / |.q.|) - s) / (1 - s))]| if ( (q `2) / |.q.| >= s & q `1 > 0 ) |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2))),((((q `2) / |.q.|) - s) / (1 + s))]| if ( (q `2) / |.q.| < s & q `1 > 0 ) otherwise q; correctness coherence ( ( (q `2) / |.q.| >= s & q `1 > 0 implies |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2))),((((q `2) / |.q.|) - s) / (1 - s))]| is Point of (TOP-REAL 2) ) & ( (q `2) / |.q.| < s & q `1 > 0 implies |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2))),((((q `2) / |.q.|) - s) / (1 + s))]| is Point of (TOP-REAL 2) ) & ( ( not (q `2) / |.q.| >= s or not q `1 > 0 ) & ( not (q `2) / |.q.| < s or not q `1 > 0 ) implies q is Point of (TOP-REAL 2) ) ); consistency for b1 being Point of (TOP-REAL 2) st (q `2) / |.q.| >= s & q `1 > 0 & (q `2) / |.q.| < s & q `1 > 0 holds ( b1 = |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2))),((((q `2) / |.q.|) - s) / (1 - s))]| iff b1 = |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2))),((((q `2) / |.q.|) - s) / (1 + s))]| ); ; end; :: deftheorem Def6 defines FanE JGRAPH_4:def_6_:_ for s being real number for q being Point of (TOP-REAL 2) holds ( ( (q `2) / |.q.| >= s & q `1 > 0 implies FanE (s,q) = |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2))),((((q `2) / |.q.|) - s) / (1 - s))]| ) & ( (q `2) / |.q.| < s & q `1 > 0 implies FanE (s,q) = |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2))),((((q `2) / |.q.|) - s) / (1 + s))]| ) & ( ( not (q `2) / |.q.| >= s or not q `1 > 0 ) & ( not (q `2) / |.q.| < s or not q `1 > 0 ) implies FanE (s,q) = q ) ); definition let s be real number ; funcs -FanMorphE -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def7: :: JGRAPH_4:def 7 for q being Point of (TOP-REAL 2) holds it . q = FanE (s,q); existence ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st for q being Point of (TOP-REAL 2) holds b1 . q = FanE (s,q) proof deffunc H1( Point of (TOP-REAL 2)) -> Point of (TOP-REAL 2) = FanE (s,$1); thus ex IT being Function of (TOP-REAL 2),(TOP-REAL 2) st for q being Point of (TOP-REAL 2) holds IT . q = H1(q) from FUNCT_2:sch_4(); ::_thesis: verum end; uniqueness for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for q being Point of (TOP-REAL 2) holds b1 . q = FanE (s,q) ) & ( for q being Point of (TOP-REAL 2) holds b2 . q = FanE (s,q) ) holds b1 = b2 proof deffunc H1( Point of (TOP-REAL 2)) -> Point of (TOP-REAL 2) = FanE (s,$1); thus for a, b being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for q being Point of (TOP-REAL 2) holds a . q = H1(q) ) & ( for q being Point of (TOP-REAL 2) holds b . q = H1(q) ) holds a = b from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def7 defines -FanMorphE JGRAPH_4:def_7_:_ for s being real number for b2 being Function of (TOP-REAL 2),(TOP-REAL 2) holds ( b2 = s -FanMorphE iff for q being Point of (TOP-REAL 2) holds b2 . q = FanE (s,q) ); theorem Th82: :: JGRAPH_4:82 for q being Point of (TOP-REAL 2) for sn being real number holds ( ( (q `2) / |.q.| >= sn & q `1 > 0 implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 <= 0 implies (sn -FanMorphE) . q = q ) ) proof let q be Point of (TOP-REAL 2); ::_thesis: for sn being real number holds ( ( (q `2) / |.q.| >= sn & q `1 > 0 implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 <= 0 implies (sn -FanMorphE) . q = q ) ) let sn be real number ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 > 0 implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 <= 0 implies (sn -FanMorphE) . q = q ) ) hereby ::_thesis: ( q `1 <= 0 implies (sn -FanMorphE) . q = q ) assume ( (q `2) / |.q.| >= sn & q `1 > 0 ) ; ::_thesis: (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| then FanE (sn,q) = |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))),((((q `2) / |.q.|) - sn) / (1 - sn))]| by Def6 .= |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by EUCLID:58 ; hence (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by Def7; ::_thesis: verum end; assume A1: q `1 <= 0 ; ::_thesis: (sn -FanMorphE) . q = q (sn -FanMorphE) . q = FanE (sn,q) by Def7; hence (sn -FanMorphE) . q = q by A1, Def6; ::_thesis: verum end; theorem Th83: :: JGRAPH_4:83 for q being Point of (TOP-REAL 2) for sn being Real st (q `2) / |.q.| <= sn & q `1 > 0 holds (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| proof let q be Point of (TOP-REAL 2); ::_thesis: for sn being Real st (q `2) / |.q.| <= sn & q `1 > 0 holds (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| let sn be Real; ::_thesis: ( (q `2) / |.q.| <= sn & q `1 > 0 implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) assume that A1: (q `2) / |.q.| <= sn and A2: q `1 > 0 ; ::_thesis: (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| now__::_thesis:_(_(_(q_`2)_/_|.q.|_<_sn_&_(sn_-FanMorphE)_._q_=_|[(|.q.|_*_(sqrt_(1_-_(((((q_`2)_/_|.q.|)_-_sn)_/_(1_+_sn))_^2)))),(|.q.|_*_((((q_`2)_/_|.q.|)_-_sn)_/_(1_+_sn)))]|_)_or_(_(q_`2)_/_|.q.|_=_sn_&_(sn_-FanMorphE)_._q_=_|[(|.q.|_*_(sqrt_(1_-_(((((q_`2)_/_|.q.|)_-_sn)_/_(1_+_sn))_^2)))),(|.q.|_*_((((q_`2)_/_|.q.|)_-_sn)_/_(1_+_sn)))]|_)_) percases ( (q `2) / |.q.| < sn or (q `2) / |.q.| = sn ) by A1, XXREAL_0:1; case (q `2) / |.q.| < sn ; ::_thesis: (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| then FanE (sn,q) = |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))),((((q `2) / |.q.|) - sn) / (1 + sn))]| by A2, Def6 .= |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by EUCLID:58 ; hence (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by Def7; ::_thesis: verum end; caseA3: (q `2) / |.q.| = sn ; ::_thesis: (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| then (((q `2) / |.q.|) - sn) / (1 - sn) = 0 ; hence (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A2, A3, Th82; ::_thesis: verum end; end; end; hence (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ; ::_thesis: verum end; theorem Th84: :: JGRAPH_4:84 for q being Point of (TOP-REAL 2) for sn being Real st - 1 < sn & sn < 1 holds ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) proof let q be Point of (TOP-REAL 2); ::_thesis: for sn being Real st - 1 < sn & sn < 1 holds ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) ) assume that A1: - 1 < sn and A2: sn < 1 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) percases ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) ) or ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) ) or q `1 < 0 or q = 0. (TOP-REAL 2) ) ; supposeA3: ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) percases ( q `1 > 0 or q `1 <= 0 ) ; supposeA4: q `1 > 0 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) then FanE (sn,q) = |.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))),((((q `2) / |.q.|) - sn) / (1 - sn))]| by A3, Def6 .= |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by EUCLID:58 ; hence ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) by A4, Def7, Th83; ::_thesis: verum end; supposeA5: q `1 <= 0 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) then A6: (sn -FanMorphE) . q = q by Th82; A7: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; A8: 1 - sn > 0 by A2, XREAL_1:149; A9: q `1 = 0 by A3, A5; |.q.| <> 0 by A3, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `2) ^2) / (|.q.| ^2) = 1 ^2 by A7, A9, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 = 1 ^2 by XCMPLX_1:76; then A10: sqrt (((q `2) / |.q.|) ^2) = 1 by SQUARE_1:22; A11: now__::_thesis:_not_q_`2_<_0 assume q `2 < 0 ; ::_thesis: contradiction then - ((q `2) / |.q.|) = 1 by A10, SQUARE_1:23; hence contradiction by A1, A3; ::_thesis: verum end; sqrt (|.q.| ^2) = |.q.| by SQUARE_1:22; then A12: |.q.| = q `2 by A7, A9, A11, SQUARE_1:22; then 1 = (q `2) / |.q.| by A3, TOPRNS_1:24, XCMPLX_1:60; then (((q `2) / |.q.|) - sn) / (1 - sn) = 1 by A8, XCMPLX_1:60; hence ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) by A2, A6, A9, A12, EUCLID:53, SQUARE_1:17, TOPRNS_1:24, XCMPLX_1:60; ::_thesis: verum end; end; end; supposeA13: ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) percases ( q `1 > 0 or q `1 <= 0 ) ; suppose q `1 > 0 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) hence ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) by Th82, Th83; ::_thesis: verum end; supposeA14: q `1 <= 0 ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) then A15: q `1 = 0 by A13; A16: 1 + sn > 0 by A1, XREAL_1:148; A17: |.q.| <> 0 by A13, TOPRNS_1:24; 1 > (q `2) / |.q.| by A2, A13, XXREAL_0:2; then 1 * |.q.| > ((q `2) / |.q.|) * |.q.| by A17, XREAL_1:68; then A18: ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & |.q.| > q `2 ) by A13, JGRAPH_3:1, TOPRNS_1:24, XCMPLX_1:87; then A19: |.q.| = - (q `2) by A15, SQUARE_1:40; A20: q `2 = - |.q.| by A15, A18, SQUARE_1:40; then - 1 = (q `2) / |.q.| by A13, TOPRNS_1:24, XCMPLX_1:197; then (((q `2) / |.q.|) - sn) / (1 + sn) = (- (1 + sn)) / (1 + sn) .= - 1 by A16, XCMPLX_1:197 ; then |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| = q by A15, A19, EUCLID:53, SQUARE_1:17; hence ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) by A1, A14, A17, A20, Th82, XCMPLX_1:197; ::_thesis: verum end; end; end; suppose ( q `1 < 0 or q = 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) hence ( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) ) ; ::_thesis: verum end; end; end; theorem Th85: :: JGRAPH_4:85 for sn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let sn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = sn; set b = 1 - sn; reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm3; assume that A1: sn < 1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 - sn > 0 by A1, XREAL_1:149; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (((r1 / r2) - sn) / (1 - sn)) and A6: g3 is continuous by A4, Th5; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in K1 by A7, A8, A9, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( proj2 . r = r `2 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_6; A11: ( g2 . s = proj2 . s & g1 . s = (2 NormF) . s ) by Lm3, Lm5; f . r = |.r.| * ((((r `2) / |.r.|) - sn) / (1 - sn)) by A2, A9; hence f . x = g3 . x by A5, A11, A10; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th86: :: JGRAPH_4:86 for sn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let sn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = sn; set b = 1 + sn; reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm3; assume that A1: - 1 < sn and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 + sn > 0 by A1, XREAL_1:148; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (((r1 / r2) - sn) / (1 + sn)) and A6: g3 is continuous by A4, Th5; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; A8: for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in dom g3 by A7, A9; then x in K1 by A7, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( proj2 . r = r `2 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_6; A11: ( g2 . s = proj2 . s & g1 . s = (2 NormF) . s ) by Lm3, Lm5; f . r = |.r.| * ((((r `2) / |.r.|) - sn) / (1 + sn)) by A2, A9; hence f . x = g3 . x by A5, A11, A10; ::_thesis: verum end; dom f = dom g3 by A7, FUNCT_2:def_1; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th87: :: JGRAPH_4:87 for sn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let sn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = sn; set b = 1 - sn; reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm3; assume that A1: sn < 1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 - sn > 0 by A1, XREAL_1:149; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (sqrt (abs (1 - ((((r1 / r2) - sn) / (1 - sn)) ^2)))) and A6: g3 is continuous by A4, Th10; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) A9: 1 - sn > 0 by A1, XREAL_1:149; assume A10: x in dom f ; ::_thesis: f . x = g3 . x then x in K1 by A7, A8, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A11: |.r.| <> 0 by A3, A10, TOPRNS_1:24; |.r.| ^2 = ((r `1) ^2) + ((r `2) ^2) by JGRAPH_3:1; then A12: ((r `2) - |.r.|) * ((r `2) + |.r.|) = - ((r `1) ^2) ; (r `1) ^2 >= 0 by XREAL_1:63; then r `2 <= |.r.| by A12, XREAL_1:93; then (r `2) / |.r.| <= |.r.| / |.r.| by XREAL_1:72; then (r `2) / |.r.| <= 1 by A11, XCMPLX_1:60; then A13: ((r `2) / |.r.|) - sn <= 1 - sn by XREAL_1:9; reconsider s = x as Point of ((TOP-REAL 2) | K1) by A10; A14: now__::_thesis:_not_(1_-_sn)_^2_=_0 assume (1 - sn) ^2 = 0 ; ::_thesis: contradiction then (1 - sn) + sn = 0 + sn by XCMPLX_1:6; hence contradiction by A1; ::_thesis: verum end; sn - ((r `2) / |.r.|) <= 0 by A3, A10, XREAL_1:47; then - (sn - ((r `2) / |.r.|)) >= - (1 - sn) by A9, XREAL_1:24; then ( (1 - sn) ^2 >= 0 & (((r `2) / |.r.|) - sn) ^2 <= (1 - sn) ^2 ) by A13, SQUARE_1:49, XREAL_1:63; then ((((r `2) / |.r.|) - sn) ^2) / ((1 - sn) ^2) <= ((1 - sn) ^2) / ((1 - sn) ^2) by XREAL_1:72; then ((((r `2) / |.r.|) - sn) ^2) / ((1 - sn) ^2) <= 1 by A14, XCMPLX_1:60; then ((((r `2) / |.r.|) - sn) / (1 - sn)) ^2 <= 1 by XCMPLX_1:76; then 1 - (((((r `2) / |.r.|) - sn) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then abs (1 - (((((r `2) / |.r.|) - sn) / (1 - sn)) ^2)) = 1 - (((((r `2) / |.r.|) - sn) / (1 - sn)) ^2) by ABSVALUE:def_1; then A15: f . r = |.r.| * (sqrt (abs (1 - (((((r `2) / |.r.|) - sn) / (1 - sn)) ^2)))) by A2, A10; A16: ( proj2 . r = r `2 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_6; ( g2 . s = proj2 . s & g1 . s = (2 NormF) . s ) by Lm3, Lm5; hence f . x = g3 . x by A5, A15, A16; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th88: :: JGRAPH_4:88 for sn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let sn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = sn; set b = 1 + sn; reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm3; assume that A1: - 1 < sn and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous A4: 1 + sn > 0 by A1, XREAL_1:148; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (sqrt (abs (1 - ((((r1 / r2) - sn) / (1 + sn)) ^2)))) and A6: g3 is continuous by A4, Th10; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then x in K1 by A7, A8, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; reconsider s = x as Point of ((TOP-REAL 2) | K1) by A9; A10: (1 + sn) ^2 > 0 by A4, SQUARE_1:12; A11: |.r.| <> 0 by A3, A9, TOPRNS_1:24; |.r.| ^2 = ((r `1) ^2) + ((r `2) ^2) by JGRAPH_3:1; then A12: ((r `2) - |.r.|) * ((r `2) + |.r.|) = - ((r `1) ^2) ; (r `1) ^2 >= 0 by XREAL_1:63; then - |.r.| <= r `2 by A12, XREAL_1:93; then (r `2) / |.r.| >= (- |.r.|) / |.r.| by XREAL_1:72; then (r `2) / |.r.| >= - 1 by A11, XCMPLX_1:197; then ((r `2) / |.r.|) - sn >= (- 1) - sn by XREAL_1:9; then A13: ((r `2) / |.r.|) - sn >= - (1 + sn) ; sn - ((r `2) / |.r.|) >= 0 by A3, A9, XREAL_1:48; then - (sn - ((r `2) / |.r.|)) <= - 0 ; then (((r `2) / |.r.|) - sn) ^2 <= (1 + sn) ^2 by A4, A13, SQUARE_1:49; then ((((r `2) / |.r.|) - sn) ^2) / ((1 + sn) ^2) <= ((1 + sn) ^2) / ((1 + sn) ^2) by A4, XREAL_1:72; then ((((r `2) / |.r.|) - sn) ^2) / ((1 + sn) ^2) <= 1 by A10, XCMPLX_1:60; then ((((r `2) / |.r.|) - sn) / (1 + sn)) ^2 <= 1 by XCMPLX_1:76; then 1 - (((((r `2) / |.r.|) - sn) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then abs (1 - (((((r `2) / |.r.|) - sn) / (1 + sn)) ^2)) = 1 - (((((r `2) / |.r.|) - sn) / (1 + sn)) ^2) by ABSVALUE:def_1; then A14: f . r = |.r.| * (sqrt (abs (1 - (((((r `2) / |.r.|) - sn) / (1 + sn)) ^2)))) by A2, A9; A15: ( proj2 . r = r `2 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_6; ( g2 . s = proj2 . s & g1 . s = (2 NormF) . s ) by Lm3, Lm5; hence f . x = g3 . x by A5, A14, A15; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th89: :: JGRAPH_4:89 for sn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set cn = sqrt (1 - (sn ^2)); set p0 = |[(sqrt (1 - (sn ^2))),sn]|; A1: |[(sqrt (1 - (sn ^2))),sn]| `1 = sqrt (1 - (sn ^2)) by EUCLID:52; |[(sqrt (1 - (sn ^2))),sn]| `2 = sn by EUCLID:52; then A2: |.|[(sqrt (1 - (sn ^2))),sn]|.| = sqrt (((sqrt (1 - (sn ^2))) ^2) + (sn ^2)) by A1, JGRAPH_3:1; assume A3: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then sn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (sn ^2) > 0 by XREAL_1:50; then A5: - (- (sqrt (1 - (sn ^2)))) > 0 by SQUARE_1:25; (sqrt (1 - (sn ^2))) ^2 = 1 - (sn ^2) by A4, SQUARE_1:def_2; then (|[(sqrt (1 - (sn ^2))),sn]| `2) / |.|[(sqrt (1 - (sn ^2))),sn]|.| = sn by A2, EUCLID:52, SQUARE_1:18; then A6: |[(sqrt (1 - (sn ^2))),sn]| in K0 by A3, A1, A5, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A7: rng (proj1 * ((sn -FanMorphE) | K1)) c= the carrier of R^1 by TOPMETR:17; A8: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & (p8 `2) / |.p8.| >= sn & p8 `1 >= 0 & p8 <> 0. (TOP-REAL 2) ) by A3; hence x in B0 by A3; ::_thesis: verum end; A9: dom ((sn -FanMorphE) | K1) c= dom (proj2 * ((sn -FanMorphE) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((sn -FanMorphE) | K1) or x in dom (proj2 * ((sn -FanMorphE) | K1)) ) assume A10: x in dom ((sn -FanMorphE) | K1) ; ::_thesis: x in dom (proj2 * ((sn -FanMorphE) | K1)) then x in (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61; then x in dom (sn -FanMorphE) by XBOOLE_0:def_4; then A11: ( dom proj2 = the carrier of (TOP-REAL 2) & (sn -FanMorphE) . x in rng (sn -FanMorphE) ) by FUNCT_1:3, FUNCT_2:def_1; ((sn -FanMorphE) | K1) . x = (sn -FanMorphE) . x by A10, FUNCT_1:47; hence x in dom (proj2 * ((sn -FanMorphE) | K1)) by A10, A11, FUNCT_1:11; ::_thesis: verum end; A12: rng (proj2 * ((sn -FanMorphE) | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj2 * ((sn -FanMorphE) | K1)) c= dom ((sn -FanMorphE) | K1) by RELAT_1:25; then dom (proj2 * ((sn -FanMorphE) | K1)) = dom ((sn -FanMorphE) | K1) by A9, XBOOLE_0:def_10 .= (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g2 = proj2 * ((sn -FanMorphE) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A12, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) A13: dom ((sn -FanMorphE) | K1) = (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A14: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A15: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A14; then A16: (sn -FanMorphE) . p = |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A3, Th84; ((sn -FanMorphE) | K1) . p = (sn -FanMorphE) . p by A15, A14, FUNCT_1:49; then g2 . p = proj2 . |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A15, A13, A14, A16, FUNCT_1:13 .= |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| `2 by PSCOMP_1:def_6 .= |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) by EUCLID:52 ; hence g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A17: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ; A18: dom ((sn -FanMorphE) | K1) c= dom (proj1 * ((sn -FanMorphE) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((sn -FanMorphE) | K1) or x in dom (proj1 * ((sn -FanMorphE) | K1)) ) assume A19: x in dom ((sn -FanMorphE) | K1) ; ::_thesis: x in dom (proj1 * ((sn -FanMorphE) | K1)) then x in (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61; then x in dom (sn -FanMorphE) by XBOOLE_0:def_4; then A20: ( dom proj1 = the carrier of (TOP-REAL 2) & (sn -FanMorphE) . x in rng (sn -FanMorphE) ) by FUNCT_1:3, FUNCT_2:def_1; ((sn -FanMorphE) | K1) . x = (sn -FanMorphE) . x by A19, FUNCT_1:47; hence x in dom (proj1 * ((sn -FanMorphE) | K1)) by A19, A20, FUNCT_1:11; ::_thesis: verum end; dom (proj1 * ((sn -FanMorphE) | K1)) c= dom ((sn -FanMorphE) | K1) by RELAT_1:25; then dom (proj1 * ((sn -FanMorphE) | K1)) = dom ((sn -FanMorphE) | K1) by A18, XBOOLE_0:def_10 .= (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g1 = proj1 * ((sn -FanMorphE) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A7, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) ) A21: dom ((sn -FanMorphE) | K1) = (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A22: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A23: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A22; then A24: (sn -FanMorphE) . p = |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A3, Th84; ((sn -FanMorphE) | K1) . p = (sn -FanMorphE) . p by A23, A22, FUNCT_1:49; then g1 . p = proj1 . |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A23, A21, A22, A24, FUNCT_1:13 .= |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| `1 by PSCOMP_1:def_5 .= |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) by EUCLID:52 ; hence g1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A25: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) ; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) A26: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A26; hence ( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then A27: f1 is continuous by A3, A25, Th87; A28: for x, y, r, s being real number st |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| proof let x, y, r, s be real number ; ::_thesis: ( |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| implies f . |[x,y]| = |[r,s]| ) assume that A29: |[x,y]| in K1 and A30: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[r,s]| set p99 = |[x,y]|; A31: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A29; A32: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then A33: f1 . |[x,y]| = |.|[x,y]|.| * (sqrt (1 - (((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 - sn)) ^2))) by A25, A29; ((sn -FanMorphE) | K0) . |[x,y]| = (sn -FanMorphE) . |[x,y]| by A29, FUNCT_1:49 .= |[(|.|[x,y]|.| * (sqrt (1 - (((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 - sn)) ^2)))),(|.|[x,y]|.| * ((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 - sn)))]| by A3, A31, Th84 .= |[r,s]| by A17, A29, A30, A32, A33 ; hence f . |[x,y]| = |[r,s]| by A3; ::_thesis: verum end; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) A34: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `2) / |.p3.| >= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A34; hence ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then f2 is continuous by A3, A17, Th85; hence f is continuous by A6, A8, A27, A28, JGRAPH_2:35; ::_thesis: verum end; theorem Th90: :: JGRAPH_4:90 for sn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set cn = sqrt (1 - (sn ^2)); set p0 = |[(sqrt (1 - (sn ^2))),sn]|; A1: |[(sqrt (1 - (sn ^2))),sn]| `1 = sqrt (1 - (sn ^2)) by EUCLID:52; |[(sqrt (1 - (sn ^2))),sn]| `2 = sn by EUCLID:52; then A2: |.|[(sqrt (1 - (sn ^2))),sn]|.| = sqrt (((sqrt (1 - (sn ^2))) ^2) + (sn ^2)) by A1, JGRAPH_3:1; assume A3: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then sn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (sn ^2) > 0 by XREAL_1:50; then A5: - (- (sqrt (1 - (sn ^2)))) > 0 by SQUARE_1:25; (sqrt (1 - (sn ^2))) ^2 = 1 - (sn ^2) by A4, SQUARE_1:def_2; then (|[(sqrt (1 - (sn ^2))),sn]| `2) / |.|[(sqrt (1 - (sn ^2))),sn]|.| = sn by A2, EUCLID:52, SQUARE_1:18; then A6: |[(sqrt (1 - (sn ^2))),sn]| in K0 by A3, A1, A5, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A7: rng (proj1 * ((sn -FanMorphE) | K1)) c= the carrier of R^1 by TOPMETR:17; A8: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & (p8 `2) / |.p8.| <= sn & p8 `1 >= 0 & p8 <> 0. (TOP-REAL 2) ) by A3; hence x in B0 by A3; ::_thesis: verum end; A9: dom ((sn -FanMorphE) | K1) c= dom (proj2 * ((sn -FanMorphE) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((sn -FanMorphE) | K1) or x in dom (proj2 * ((sn -FanMorphE) | K1)) ) assume A10: x in dom ((sn -FanMorphE) | K1) ; ::_thesis: x in dom (proj2 * ((sn -FanMorphE) | K1)) then x in (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61; then x in dom (sn -FanMorphE) by XBOOLE_0:def_4; then A11: ( dom proj2 = the carrier of (TOP-REAL 2) & (sn -FanMorphE) . x in rng (sn -FanMorphE) ) by FUNCT_1:3, FUNCT_2:def_1; ((sn -FanMorphE) | K1) . x = (sn -FanMorphE) . x by A10, FUNCT_1:47; hence x in dom (proj2 * ((sn -FanMorphE) | K1)) by A10, A11, FUNCT_1:11; ::_thesis: verum end; A12: rng (proj2 * ((sn -FanMorphE) | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj2 * ((sn -FanMorphE) | K1)) c= dom ((sn -FanMorphE) | K1) by RELAT_1:25; then dom (proj2 * ((sn -FanMorphE) | K1)) = dom ((sn -FanMorphE) | K1) by A9, XBOOLE_0:def_10 .= (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g2 = proj2 * ((sn -FanMorphE) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A12, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) A13: dom ((sn -FanMorphE) | K1) = (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A14: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A15: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A14; then A16: (sn -FanMorphE) . p = |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A3, Th84; ((sn -FanMorphE) | K1) . p = (sn -FanMorphE) . p by A15, A14, FUNCT_1:49; then g2 . p = proj2 . |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A15, A13, A14, A16, FUNCT_1:13 .= |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| `2 by PSCOMP_1:def_6 .= |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) by EUCLID:52 ; hence g2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A17: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ; A18: dom ((sn -FanMorphE) | K1) c= dom (proj1 * ((sn -FanMorphE) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((sn -FanMorphE) | K1) or x in dom (proj1 * ((sn -FanMorphE) | K1)) ) assume A19: x in dom ((sn -FanMorphE) | K1) ; ::_thesis: x in dom (proj1 * ((sn -FanMorphE) | K1)) then x in (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61; then x in dom (sn -FanMorphE) by XBOOLE_0:def_4; then A20: ( dom proj1 = the carrier of (TOP-REAL 2) & (sn -FanMorphE) . x in rng (sn -FanMorphE) ) by FUNCT_1:3, FUNCT_2:def_1; ((sn -FanMorphE) | K1) . x = (sn -FanMorphE) . x by A19, FUNCT_1:47; hence x in dom (proj1 * ((sn -FanMorphE) | K1)) by A19, A20, FUNCT_1:11; ::_thesis: verum end; dom (proj1 * ((sn -FanMorphE) | K1)) c= dom ((sn -FanMorphE) | K1) by RELAT_1:25; then dom (proj1 * ((sn -FanMorphE) | K1)) = dom ((sn -FanMorphE) | K1) by A18, XBOOLE_0:def_10 .= (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g1 = proj1 * ((sn -FanMorphE) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A7, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) ) A21: dom ((sn -FanMorphE) | K1) = (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A22: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A23: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A22; then A24: (sn -FanMorphE) . p = |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A3, Th84; ((sn -FanMorphE) | K1) . p = (sn -FanMorphE) . p by A23, A22, FUNCT_1:49; then g1 . p = proj1 . |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A23, A21, A22, A24, FUNCT_1:13 .= |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| `1 by PSCOMP_1:def_5 .= |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) by EUCLID:52 ; hence g1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A25: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) ; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) A26: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A26; hence ( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then A27: f1 is continuous by A3, A25, Th88; A28: for x, y, r, s being real number st |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds f . |[x,y]| = |[r,s]| proof let x, y, r, s be real number ; ::_thesis: ( |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| implies f . |[x,y]| = |[r,s]| ) assume that A29: |[x,y]| in K1 and A30: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[r,s]| set p99 = |[x,y]|; A31: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A29; A32: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then A33: f1 . |[x,y]| = |.|[x,y]|.| * (sqrt (1 - (((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 + sn)) ^2))) by A25, A29; ((sn -FanMorphE) | K0) . |[x,y]| = (sn -FanMorphE) . |[x,y]| by A29, FUNCT_1:49 .= |[(|.|[x,y]|.| * (sqrt (1 - (((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 + sn)) ^2)))),(|.|[x,y]|.| * ((((|[x,y]| `2) / |.|[x,y]|.|) - sn) / (1 + sn)))]| by A3, A31, Th84 .= |[r,s]| by A17, A29, A30, A32, A33 ; hence f . |[x,y]| = |[r,s]| by A3; ::_thesis: verum end; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) A34: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `2) / |.p3.| <= sn & p3 `1 >= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A34; hence ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then f2 is continuous by A3, A17, Th86; hence f is continuous by A6, A8, A27, A28, JGRAPH_2:35; ::_thesis: verum end; theorem Th91: :: JGRAPH_4:91 for sn being Real for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 >= 0 ) } holds K03 is closed proof defpred S1[ Point of (TOP-REAL 2)] means $1 `1 >= 0 ; let sn be Real; ::_thesis: for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 >= 0 ) } holds K03 is closed let K003 be Subset of (TOP-REAL 2); ::_thesis: ( K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 >= 0 ) } implies K003 is closed ) assume A1: K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 >= 0 ) } ; ::_thesis: K003 is closed reconsider KX = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); defpred S2[ Point of (TOP-REAL 2)] means $1 `2 >= sn * |.$1.|; reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A2: { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ { p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } from DOMAIN_1:sch_10(); ( K1 is closed & KX is closed ) by Lm7, JORDAN6:4; hence K003 is closed by A1, A2, TOPS_1:8; ::_thesis: verum end; theorem Th92: :: JGRAPH_4:92 for sn being Real for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 >= 0 ) } holds K03 is closed proof defpred S1[ Point of (TOP-REAL 2)] means $1 `1 >= 0 ; let sn be Real; ::_thesis: for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 >= 0 ) } holds K03 is closed let K003 be Subset of (TOP-REAL 2); ::_thesis: ( K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 >= 0 ) } implies K003 is closed ) assume A1: K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 >= 0 ) } ; ::_thesis: K003 is closed reconsider KX = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); defpred S2[ Point of (TOP-REAL 2)] means $1 `2 <= sn * |.$1.|; reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A2: { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ { p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } from DOMAIN_1:sch_10(); ( K1 is closed & KX is closed ) by Lm9, JORDAN6:4; hence K003 is closed by A1, A2, TOPS_1:8; ::_thesis: verum end; theorem Th93: :: JGRAPH_4:93 for sn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set cn = sqrt (1 - (sn ^2)); set p0 = |[(sqrt (1 - (sn ^2))),sn]|; A1: |[(sqrt (1 - (sn ^2))),sn]| `1 = sqrt (1 - (sn ^2)) by EUCLID:52; defpred S1[ Point of (TOP-REAL 2)] means ( ($1 `2) / |.$1.| >= sn & $1 `1 >= 0 & $1 <> 0. (TOP-REAL 2) ); |[(sqrt (1 - (sn ^2))),sn]| `2 = sn by EUCLID:52; then A2: |.|[(sqrt (1 - (sn ^2))),sn]|.| = sqrt (((sqrt (1 - (sn ^2))) ^2) + (sn ^2)) by A1, JGRAPH_3:1; assume A3: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then sn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (sn ^2) > 0 by XREAL_1:50; then A5: |[(sqrt (1 - (sn ^2))),sn]| `1 > 0 by A1, SQUARE_1:25; then |[(sqrt (1 - (sn ^2))),sn]| in K0 by A3, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; (sqrt (1 - (sn ^2))) ^2 = 1 - (sn ^2) by A4, SQUARE_1:def_2; then A6: (|[(sqrt (1 - (sn ^2))),sn]| `2) / |.|[(sqrt (1 - (sn ^2))),sn]|.| = sn by A2, EUCLID:52, SQUARE_1:18; then A7: |[(sqrt (1 - (sn ^2))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } by A5, JGRAPH_2:3; { p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K001 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A7; A8: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; defpred S2[ Point of (TOP-REAL 2)] means ( $1 `2 >= sn * |.$1.| & $1 `1 >= 0 ); { p where p is Point of (TOP-REAL 2) : S2[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K003 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 >= 0 ) } as Subset of (TOP-REAL 2) ; defpred S3[ Point of (TOP-REAL 2)] means ( ($1 `2) / |.$1.| <= sn & $1 `1 >= 0 & $1 <> 0. (TOP-REAL 2) ); A9: { p where p is Point of (TOP-REAL 2) : S3[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); A10: - (- (sqrt (1 - (sn ^2)))) > 0 by A4, SQUARE_1:25; then |[(sqrt (1 - (sn ^2))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, JGRAPH_2:3; then reconsider K111 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A9; A11: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def_5; A12: rng ((sn -FanMorphE) | K001) c= K1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((sn -FanMorphE) | K001) or y in K1 ) assume y in rng ((sn -FanMorphE) | K001) ; ::_thesis: y in K1 then consider x being set such that A13: x in dom ((sn -FanMorphE) | K001) and A14: y = ((sn -FanMorphE) | K001) . x by FUNCT_1:def_3; x in dom (sn -FanMorphE) by A13, RELAT_1:57; then reconsider q = x as Point of (TOP-REAL 2) ; A15: y = (sn -FanMorphE) . q by A13, A14, FUNCT_1:47; dom ((sn -FanMorphE) | K001) = (dom (sn -FanMorphE)) /\ K001 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K001 by FUNCT_2:def_1 .= K001 by XBOOLE_1:28 ; then A16: ex p2 being Point of (TOP-REAL 2) st ( p2 = q & (p2 `2) / |.p2.| >= sn & p2 `1 >= 0 & p2 <> 0. (TOP-REAL 2) ) by A13; then A17: ((q `2) / |.q.|) - sn >= 0 by XREAL_1:48; |.q.| <> 0 by A16, TOPRNS_1:24; then A18: |.q.| ^2 > 0 ^2 by SQUARE_1:12; set q4 = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]|; A19: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)) by EUCLID:52; A20: 1 - sn > 0 by A3, XREAL_1:149; 0 <= (q `1) ^2 by XREAL_1:63; then 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then (q `2) ^2 <= |.q.| ^2 by JGRAPH_3:1; then ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A18, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `2) / |.q.| by SQUARE_1:51; then 1 - sn >= ((q `2) / |.q.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((q `2) / |.q.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A20, XREAL_1:72; then - 1 <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A20, XCMPLX_1:197; then ((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A20, A17, SQUARE_1:49; then A21: 1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A22: 1 - ((- ((((q `2) / |.q.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; sqrt (1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2)) >= 0 by A21, SQUARE_1:def_2; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) ^2) / ((1 - sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((q `2) / |.q.|) - sn) ^2) / ((1 - sn) ^2))) >= 0 ; then A23: sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) >= 0 by XCMPLX_1:76; A24: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) by EUCLID:52; then A25: (|[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) by A22, SQUARE_1:def_2 ; |.|[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]|.| ^2 = ((|[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `1) ^2) + ((|[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A19, A25 ; then A26: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| <> 0. (TOP-REAL 2) by A18, TOPRNS_1:23; (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by A3, A16, Th84; hence y in K1 by A3, A15, A24, A23, A26; ::_thesis: verum end; A27: { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } c= K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 ) assume x in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in K1 then ex p being Point of (TOP-REAL 2) st ( p = x & (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) ; hence x in K1 by A3; ::_thesis: verum end; |[(sqrt (1 - (sn ^2))),sn]| <> 0. (TOP-REAL 2) by A1, A4, JGRAPH_2:3, SQUARE_1:25; then not |[(sqrt (1 - (sn ^2))),sn]| in {(0. (TOP-REAL 2))} by TARSKI:def_1; then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A3, XBOOLE_0:def_5; K1 c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in D ) assume A28: x in K1 ; ::_thesis: x in D then ex p6 being Point of (TOP-REAL 2) st ( p6 = x & p6 `1 >= 0 & p6 <> 0. (TOP-REAL 2) ) by A3; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in D by A3, A28, XBOOLE_0:def_5; ::_thesis: verum end; then D = K1 \/ D by XBOOLE_1:12; then A29: (TOP-REAL 2) | K1 is SubSpace of (TOP-REAL 2) | D by TOPMETR:4; A30: { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } c= K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 ) assume x in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in K1 then ex p being Point of (TOP-REAL 2) st ( p = x & (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) ; hence x in K1 by A3; ::_thesis: verum end; then reconsider K00 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A7, PRE_TOPC:8; A31: K003 is closed by Th91; A32: rng ((sn -FanMorphE) | K111) c= K1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((sn -FanMorphE) | K111) or y in K1 ) assume y in rng ((sn -FanMorphE) | K111) ; ::_thesis: y in K1 then consider x being set such that A33: x in dom ((sn -FanMorphE) | K111) and A34: y = ((sn -FanMorphE) | K111) . x by FUNCT_1:def_3; x in dom (sn -FanMorphE) by A33, RELAT_1:57; then reconsider q = x as Point of (TOP-REAL 2) ; A35: y = (sn -FanMorphE) . q by A33, A34, FUNCT_1:47; dom ((sn -FanMorphE) | K111) = (dom (sn -FanMorphE)) /\ K111 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K111 by FUNCT_2:def_1 .= K111 by XBOOLE_1:28 ; then A36: ex p2 being Point of (TOP-REAL 2) st ( p2 = q & (p2 `2) / |.p2.| <= sn & p2 `1 >= 0 & p2 <> 0. (TOP-REAL 2) ) by A33; then A37: ((q `2) / |.q.|) - sn <= 0 by XREAL_1:47; |.q.| <> 0 by A36, TOPRNS_1:24; then A38: |.q.| ^2 > 0 ^2 by SQUARE_1:12; set q4 = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|; A39: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) by EUCLID:52; A40: 1 + sn > 0 by A3, XREAL_1:148; 0 <= (q `1) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A38, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `2) / |.q.| by SQUARE_1:51; then (- 1) - sn <= ((q `2) / |.q.|) - sn by XREAL_1:9; then (- (1 + sn)) / (1 + sn) <= (((q `2) / |.q.|) - sn) / (1 + sn) by A40, XREAL_1:72; then - 1 <= (((q `2) / |.q.|) - sn) / (1 + sn) by A40, XCMPLX_1:197; then A41: ((((q `2) / |.q.|) - sn) / (1 + sn)) ^2 <= 1 ^2 by A40, A37, SQUARE_1:49; then A42: 1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2) >= 0 by XREAL_1:48; 1 - ((- ((((q `2) / |.q.|) - sn) / (1 + sn))) ^2) >= 0 by A41, XREAL_1:48; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2) >= 0 by XCMPLX_1:187; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) ^2) / ((1 + sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((q `2) / |.q.|) - sn) ^2) / ((1 + sn) ^2))) >= 0 ; then A43: sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) >= 0 by XCMPLX_1:76; A44: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) by EUCLID:52; then A45: (|[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) by A42, SQUARE_1:def_2 ; |.|[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|.| ^2 = ((|[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `1) ^2) + ((|[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A39, A45 ; then A46: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| <> 0. (TOP-REAL 2) by A38, TOPRNS_1:23; (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A3, A36, Th84; hence y in K1 by A3, A35, A44, A43, A46; ::_thesis: verum end; the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; then A47: rng (f | K00) c= D ; the carrier of ((TOP-REAL 2) | B0) = the carrier of ((TOP-REAL 2) | D) ; then A48: dom f = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1 .= K1 by PRE_TOPC:8 ; then dom (f | K00) = K00 by A30, RELAT_1:62 .= the carrier of (((TOP-REAL 2) | K1) | K00) by PRE_TOPC:8 ; then reconsider f1 = f | K00 as Function of (((TOP-REAL 2) | K1) | K00),((TOP-REAL 2) | D) by A47, FUNCT_2:2; A49: the carrier of ((TOP-REAL 2) | K1) = K0 by PRE_TOPC:8; |[(sqrt (1 - (sn ^2))),sn]| in { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A10, A6, JGRAPH_2:3; then reconsider K11 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A27, PRE_TOPC:8; A50: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; A51: dom (sn -FanMorphE) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then dom ((sn -FanMorphE) | K001) = K001 by RELAT_1:62 .= the carrier of ((TOP-REAL 2) | K001) by PRE_TOPC:8 ; then reconsider f3 = (sn -FanMorphE) | K001 as Function of ((TOP-REAL 2) | K001),((TOP-REAL 2) | K1) by A8, A12, FUNCT_2:2; A52: D <> {} ; dom ((sn -FanMorphE) | K111) = K111 by A51, RELAT_1:62 .= the carrier of ((TOP-REAL 2) | K111) by PRE_TOPC:8 ; then reconsider f4 = (sn -FanMorphE) | K111 as Function of ((TOP-REAL 2) | K111),((TOP-REAL 2) | K1) by A50, A32, FUNCT_2:2; the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; then A53: rng (f | K11) c= D ; dom (f | K11) = K11 by A27, A48, RELAT_1:62 .= the carrier of (((TOP-REAL 2) | K1) | K11) by PRE_TOPC:8 ; then reconsider f2 = f | K11 as Function of (((TOP-REAL 2) | K1) | K11),((TOP-REAL 2) | D) by A53, FUNCT_2:2; the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then ( ((TOP-REAL 2) | K1) | K11 = (TOP-REAL 2) | K111 & f2 = f4 ) by A3, FUNCT_1:51, GOBOARD9:2; then A54: f2 is continuous by A3, A29, Th90, PRE_TOPC:26; the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then ( ((TOP-REAL 2) | K1) | K00 = (TOP-REAL 2) | K001 & f1 = f3 ) by A3, FUNCT_1:51, GOBOARD9:2; then A55: f1 is continuous by A3, A29, Th89, PRE_TOPC:26; A56: dom f2 = the carrier of (((TOP-REAL 2) | K1) | K11) by FUNCT_2:def_1 .= K11 by PRE_TOPC:8 ; defpred S4[ Point of (TOP-REAL 2)] means ( $1 `2 <= sn * |.$1.| & $1 `1 >= 0 ); { p where p is Point of (TOP-REAL 2) : S4[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K004 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 >= 0 ) } as Subset of (TOP-REAL 2) ; A57: K004 /\ K1 c= K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K004 /\ K1 or x in K11 ) assume A58: x in K004 /\ K1 ; ::_thesis: x in K11 then x in K004 by XBOOLE_0:def_4; then consider q1 being Point of (TOP-REAL 2) such that A59: q1 = x and A60: q1 `2 <= sn * |.q1.| and q1 `1 >= 0 ; x in K1 by A58, XBOOLE_0:def_4; then A61: ex q2 being Point of (TOP-REAL 2) st ( q2 = x & q2 `1 >= 0 & q2 <> 0. (TOP-REAL 2) ) by A3; (q1 `2) / |.q1.| <= (sn * |.q1.|) / |.q1.| by A60, XREAL_1:72; then (q1 `2) / |.q1.| <= sn by A59, A61, TOPRNS_1:24, XCMPLX_1:89; hence x in K11 by A59, A61; ::_thesis: verum end; A62: K004 is closed by Th92; K11 c= K004 /\ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K11 or x in K004 /\ K1 ) assume x in K11 ; ::_thesis: x in K004 /\ K1 then consider p being Point of (TOP-REAL 2) such that A63: p = x and A64: (p `2) / |.p.| <= sn and A65: p `1 >= 0 and A66: p <> 0. (TOP-REAL 2) ; ((p `2) / |.p.|) * |.p.| <= sn * |.p.| by A64, XREAL_1:64; then p `2 <= sn * |.p.| by A66, TOPRNS_1:24, XCMPLX_1:87; then A67: x in K004 by A63, A65; x in K1 by A3, A63, A65, A66; hence x in K004 /\ K1 by A67, XBOOLE_0:def_4; ::_thesis: verum end; then K11 = K004 /\ ([#] ((TOP-REAL 2) | K1)) by A11, A57, XBOOLE_0:def_10; then A68: K11 is closed by A62, PRE_TOPC:13; A69: K003 /\ K1 c= K00 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K003 /\ K1 or x in K00 ) assume A70: x in K003 /\ K1 ; ::_thesis: x in K00 then x in K003 by XBOOLE_0:def_4; then consider q1 being Point of (TOP-REAL 2) such that A71: q1 = x and A72: q1 `2 >= sn * |.q1.| and q1 `1 >= 0 ; x in K1 by A70, XBOOLE_0:def_4; then A73: ex q2 being Point of (TOP-REAL 2) st ( q2 = x & q2 `1 >= 0 & q2 <> 0. (TOP-REAL 2) ) by A3; (q1 `2) / |.q1.| >= (sn * |.q1.|) / |.q1.| by A72, XREAL_1:72; then (q1 `2) / |.q1.| >= sn by A71, A73, TOPRNS_1:24, XCMPLX_1:89; hence x in K00 by A71, A73; ::_thesis: verum end; K00 c= K003 /\ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K00 or x in K003 /\ K1 ) assume x in K00 ; ::_thesis: x in K003 /\ K1 then consider p being Point of (TOP-REAL 2) such that A74: p = x and A75: (p `2) / |.p.| >= sn and A76: p `1 >= 0 and A77: p <> 0. (TOP-REAL 2) ; ((p `2) / |.p.|) * |.p.| >= sn * |.p.| by A75, XREAL_1:64; then p `2 >= sn * |.p.| by A77, TOPRNS_1:24, XCMPLX_1:87; then A78: x in K003 by A74, A76; x in K1 by A3, A74, A76, A77; hence x in K003 /\ K1 by A78, XBOOLE_0:def_4; ::_thesis: verum end; then K00 = K003 /\ ([#] ((TOP-REAL 2) | K1)) by A11, A69, XBOOLE_0:def_10; then A79: K00 is closed by A31, PRE_TOPC:13; set T1 = ((TOP-REAL 2) | K1) | K00; set T2 = ((TOP-REAL 2) | K1) | K11; A80: [#] (((TOP-REAL 2) | K1) | K11) = K11 by PRE_TOPC:def_5; A81: [#] (((TOP-REAL 2) | K1) | K00) = K00 by PRE_TOPC:def_5; A82: for p being set st p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) holds f1 . p = f2 . p proof let p be set ; ::_thesis: ( p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) implies f1 . p = f2 . p ) assume A83: p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) ; ::_thesis: f1 . p = f2 . p then p in K00 by A81, XBOOLE_0:def_4; hence f1 . p = f . p by FUNCT_1:49 .= f2 . p by A80, A83, FUNCT_1:49 ; ::_thesis: verum end; A84: K1 c= K00 \/ K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in K00 \/ K11 ) assume x in K1 ; ::_thesis: x in K00 \/ K11 then consider p being Point of (TOP-REAL 2) such that A85: ( p = x & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) by A3; percases ( (p `2) / |.p.| >= sn or (p `2) / |.p.| < sn ) ; suppose (p `2) / |.p.| >= sn ; ::_thesis: x in K00 \/ K11 then x in K00 by A85; hence x in K00 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; suppose (p `2) / |.p.| < sn ; ::_thesis: x in K00 \/ K11 then x in K11 by A85; hence x in K00 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; end; end; then ([#] (((TOP-REAL 2) | K1) | K00)) \/ ([#] (((TOP-REAL 2) | K1) | K11)) = [#] ((TOP-REAL 2) | K1) by A81, A80, A11, XBOOLE_0:def_10; then consider h being Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) such that A86: h = f1 +* f2 and A87: h is continuous by A81, A80, A79, A68, A55, A54, A82, JGRAPH_2:1; A88: dom h = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; A89: dom f1 = the carrier of (((TOP-REAL 2) | K1) | K00) by FUNCT_2:def_1 .= K00 by PRE_TOPC:8 ; A90: for y being set st y in dom h holds h . y = f . y proof let y be set ; ::_thesis: ( y in dom h implies h . y = f . y ) assume A91: y in dom h ; ::_thesis: h . y = f . y now__::_thesis:_h_._y_=_f_._y percases ( ( y in K00 & not y in K11 ) or y in K11 ) by A84, A88, A49, A91, XBOOLE_0:def_3; supposeA92: ( y in K00 & not y in K11 ) ; ::_thesis: h . y = f . y then y in (dom f1) \/ (dom f2) by A89, XBOOLE_0:def_3; hence h . y = f1 . y by A56, A86, A92, FUNCT_4:def_1 .= f . y by A92, FUNCT_1:49 ; ::_thesis: verum end; supposeA93: y in K11 ; ::_thesis: h . y = f . y then y in (dom f1) \/ (dom f2) by A56, XBOOLE_0:def_3; hence h . y = f2 . y by A56, A86, A93, FUNCT_4:def_1 .= f . y by A93, FUNCT_1:49 ; ::_thesis: verum end; end; end; hence h . y = f . y ; ::_thesis: verum end; K0 = the carrier of ((TOP-REAL 2) | K0) by PRE_TOPC:8 .= dom f by A52, FUNCT_2:def_1 ; hence f is continuous by A87, A88, A90, FUNCT_1:2, PRE_TOPC:8; ::_thesis: verum end; theorem Th94: :: JGRAPH_4:94 for sn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set cn = sqrt (1 - (sn ^2)); set p0 = |[(- (sqrt (1 - (sn ^2)))),(- sn)]|; assume A1: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then sn ^2 < 1 ^2 by SQUARE_1:50; then 1 - (sn ^2) > 0 by XREAL_1:50; then - (- (sqrt (1 - (sn ^2)))) > 0 by SQUARE_1:25; then A2: ( |[(- (sqrt (1 - (sn ^2)))),(- sn)]| `1 = - (sqrt (1 - (sn ^2))) & - (sqrt (1 - (sn ^2))) < 0 ) by EUCLID:52; then |[(- (sqrt (1 - (sn ^2)))),(- sn)]| in K0 by A1, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; not |[(- (sqrt (1 - (sn ^2)))),(- sn)]| in {(0. (TOP-REAL 2))} by A2, JGRAPH_2:3, TARSKI:def_1; then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A1, XBOOLE_0:def_5; A3: K1 c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in D ) assume x in K1 ; ::_thesis: x in D then consider p2 being Point of (TOP-REAL 2) such that A4: p2 = x and p2 `1 <= 0 and A5: p2 <> 0. (TOP-REAL 2) by A1; not p2 in {(0. (TOP-REAL 2))} by A5, TARSKI:def_1; hence x in D by A1, A4, XBOOLE_0:def_5; ::_thesis: verum end; for p being Point of ((TOP-REAL 2) | K1) for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) proof let p be Point of ((TOP-REAL 2) | K1); ::_thesis: for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) let V be Subset of ((TOP-REAL 2) | D); ::_thesis: ( f . p in V & V is open implies ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) ) assume that A6: f . p in V and A7: V is open ; ::_thesis: ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) consider V2 being Subset of (TOP-REAL 2) such that A8: V2 is open and A9: V2 /\ ([#] ((TOP-REAL 2) | D)) = V by A7, TOPS_2:24; reconsider W2 = V2 /\ ([#] ((TOP-REAL 2) | K1)) as Subset of ((TOP-REAL 2) | K1) ; A10: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def_5; then A11: f . p = (sn -FanMorphE) . p by A1, FUNCT_1:49; A12: f .: W2 c= V proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f .: W2 or y in V ) assume y in f .: W2 ; ::_thesis: y in V then consider x being set such that A13: x in dom f and A14: x in W2 and A15: y = f . x by FUNCT_1:def_6; f is Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) ; then dom f = K1 by A10, FUNCT_2:def_1; then consider p4 being Point of (TOP-REAL 2) such that A16: x = p4 and A17: p4 `1 <= 0 and p4 <> 0. (TOP-REAL 2) by A1, A13; A18: p4 in V2 by A14, A16, XBOOLE_0:def_4; p4 in [#] ((TOP-REAL 2) | K1) by A13, A16; then p4 in D by A3, A10; then A19: p4 in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; f . p4 = (sn -FanMorphE) . p4 by A1, A10, A13, A16, FUNCT_1:49 .= p4 by A17, Th82 ; hence y in V by A9, A15, A16, A18, A19, XBOOLE_0:def_4; ::_thesis: verum end; p in the carrier of ((TOP-REAL 2) | K1) ; then consider q being Point of (TOP-REAL 2) such that A20: q = p and A21: q `1 <= 0 and q <> 0. (TOP-REAL 2) by A1, A10; (sn -FanMorphE) . q = q by A21, Th82; then p in V2 by A6, A9, A11, A20, XBOOLE_0:def_4; then A22: p in W2 by XBOOLE_0:def_4; W2 is open by A8, TOPS_2:24; hence ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) by A22, A12; ::_thesis: verum end; hence f is continuous by JGRAPH_2:10; ::_thesis: verum end; theorem Th95: :: JGRAPH_4:95 for sn being Real for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:8; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & p8 `1 >= 0 & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; hence f is continuous by A1, Th93; ::_thesis: verum end; theorem Th96: :: JGRAPH_4:96 for sn being Real for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let sn be Real; ::_thesis: for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:8; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & p8 `1 <= 0 & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; hence f is continuous by A1, Th94; ::_thesis: verum end; theorem Th97: :: JGRAPH_4:97 for sn being Real for p being Point of (TOP-REAL 2) holds |.((sn -FanMorphE) . p).| = |.p.| proof let sn be Real; ::_thesis: for p being Point of (TOP-REAL 2) holds |.((sn -FanMorphE) . p).| = |.p.| let p be Point of (TOP-REAL 2); ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| set f = sn -FanMorphE ; set z = (sn -FanMorphE) . p; reconsider q = p as Point of (TOP-REAL 2) ; reconsider qz = (sn -FanMorphE) . p as Point of (TOP-REAL 2) ; percases ( ( (q `2) / |.q.| >= sn & q `1 > 0 ) or ( (q `2) / |.q.| < sn & q `1 > 0 ) or q `1 <= 0 ) ; supposeA1: ( (q `2) / |.q.| >= sn & q `1 > 0 ) ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| then A2: (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by Th82; then A3: qz `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) by EUCLID:52; A4: qz `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)) by A2, EUCLID:52; A5: ((q `2) / |.q.|) - sn >= 0 by A1, XREAL_1:48; A6: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; |.q.| <> 0 by A1, JGRAPH_2:3, TOPRNS_1:24; then A7: |.q.| ^2 > 0 by SQUARE_1:12; 0 <= (q `1) ^2 by XREAL_1:63; then 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by A6, XREAL_1:72; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A7, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `2) / |.q.| by SQUARE_1:51; then A8: 1 - sn >= ((q `2) / |.q.|) - sn by XREAL_1:9; percases ( 1 - sn = 0 or 1 - sn <> 0 ) ; supposeA9: 1 - sn = 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| A10: (((q `2) / |.q.|) - sn) / (1 - sn) = (((q `2) / |.q.|) - sn) * ((1 - sn) ") by XCMPLX_0:def_9 .= (((q `2) / |.q.|) - sn) * 0 by A9 .= 0 ; then 1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2) = 1 ; then (sn -FanMorphE) . q = |[(|.q.| * 1),(|.q.| * 0)]| by A1, A10, Th82, SQUARE_1:18 .= |[|.q.|,0]| ; then ( ((sn -FanMorphE) . q) `1 = |.q.| & ((sn -FanMorphE) . q) `2 = 0 ) by EUCLID:52; then |.((sn -FanMorphE) . p).| = sqrt ((|.q.| ^2) + (0 ^2)) by JGRAPH_3:1 .= |.q.| by SQUARE_1:22 ; hence |.((sn -FanMorphE) . p).| = |.p.| ; ::_thesis: verum end; supposeA11: 1 - sn <> 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| percases ( 1 - sn > 0 or 1 - sn < 0 ) by A11; supposeA12: 1 - sn > 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| - (1 - sn) <= - (((q `2) / |.q.|) - sn) by A8, XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A12, XREAL_1:72; then - 1 <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A12, XCMPLX_1:197; then ((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A5, A12, SQUARE_1:49; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A13: 1 - ((- ((((q `2) / |.q.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; A14: (qz `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) ^2) by A3 .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) by A13, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A4, A14 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; hence |.((sn -FanMorphE) . p).| = |.p.| by SQUARE_1:22; ::_thesis: verum end; supposeA15: 1 - sn < 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| 0 + ((q `2) ^2) < ((q `1) ^2) + ((q `2) ^2) by A1, SQUARE_1:12, XREAL_1:8; then ((q `2) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A7, A6, XREAL_1:74; then ((q `2) ^2) / (|.q.| ^2) < 1 by A7, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 < 1 by XCMPLX_1:76; then A16: 1 > (q `2) / |.p.| by SQUARE_1:52; ((q `2) / |.q.|) - sn >= 0 by A1, XREAL_1:48; hence |.((sn -FanMorphE) . p).| = |.p.| by A15, A16, XREAL_1:9; ::_thesis: verum end; end; end; end; end; supposeA17: ( (q `2) / |.q.| < sn & q `1 > 0 ) ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| then |.q.| <> 0 by JGRAPH_2:3, TOPRNS_1:24; then A18: |.q.| ^2 > 0 by SQUARE_1:12; A19: ((q `2) / |.q.|) - sn < 0 by A17, XREAL_1:49; A20: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; 0 <= (q `1) ^2 by XREAL_1:63; then 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by A20, XREAL_1:72; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A18, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `2) / |.q.| by SQUARE_1:51; then A21: (- 1) - sn <= ((q `2) / |.q.|) - sn by XREAL_1:9; A22: (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A17, Th83; then A23: qz `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) by EUCLID:52; A24: qz `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) by A22, EUCLID:52; percases ( 1 + sn = 0 or 1 + sn <> 0 ) ; supposeA25: 1 + sn = 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| (((q `2) / |.q.|) - sn) / (1 + sn) = (((q `2) / |.q.|) - sn) * ((1 + sn) ") by XCMPLX_0:def_9 .= (((q `2) / |.q.|) - sn) * 0 by A25 .= 0 ; then ( ((sn -FanMorphE) . q) `1 = |.q.| & ((sn -FanMorphE) . q) `2 = 0 ) by A22, EUCLID:52, SQUARE_1:18; then |.((sn -FanMorphE) . p).| = sqrt ((|.q.| ^2) + (0 ^2)) by JGRAPH_3:1 .= |.q.| by SQUARE_1:22 ; hence |.((sn -FanMorphE) . p).| = |.p.| ; ::_thesis: verum end; supposeA26: 1 + sn <> 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| percases ( 1 + sn > 0 or 1 + sn < 0 ) by A26; supposeA27: 1 + sn > 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| then (- (1 + sn)) / (1 + sn) <= (((q `2) / |.q.|) - sn) / (1 + sn) by A21, XREAL_1:72; then - 1 <= (((q `2) / |.q.|) - sn) / (1 + sn) by A27, XCMPLX_1:197; then ((((q `2) / |.q.|) - sn) / (1 + sn)) ^2 <= 1 ^2 by A19, A27, SQUARE_1:49; then A28: 1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2) >= 0 by XREAL_1:48; A29: (qz `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) ^2) by A23 .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) by A28, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A24, A29 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; hence |.((sn -FanMorphE) . p).| = |.p.| by SQUARE_1:22; ::_thesis: verum end; supposeA30: 1 + sn < 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| 0 + ((q `2) ^2) < ((q `1) ^2) + ((q `2) ^2) by A17, SQUARE_1:12, XREAL_1:8; then ((q `2) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A18, A20, XREAL_1:74; then ((q `2) ^2) / (|.q.| ^2) < 1 by A18, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (q `2) / |.p.| by SQUARE_1:52; then A31: ((q `2) / |.q.|) - sn > (- 1) - sn by XREAL_1:9; - (1 + sn) > - 0 by A30, XREAL_1:24; hence |.((sn -FanMorphE) . p).| = |.p.| by A17, A31, XREAL_1:49; ::_thesis: verum end; end; end; end; end; suppose q `1 <= 0 ; ::_thesis: |.((sn -FanMorphE) . p).| = |.p.| hence |.((sn -FanMorphE) . p).| = |.p.| by Th82; ::_thesis: verum end; end; end; theorem Th98: :: JGRAPH_4:98 for sn being Real for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds (sn -FanMorphE) . x in K0 proof let sn be Real; ::_thesis: for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds (sn -FanMorphE) . x in K0 let x, K0 be set ; ::_thesis: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } implies (sn -FanMorphE) . x in K0 ) assume A1: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: (sn -FanMorphE) . x in K0 then consider p being Point of (TOP-REAL 2) such that A2: p = x and A3: p `1 >= 0 and A4: p <> 0. (TOP-REAL 2) ; A5: now__::_thesis:_not_|.p.|_<=_0 assume |.p.| <= 0 ; ::_thesis: contradiction then |.p.| = 0 ; hence contradiction by A4, TOPRNS_1:24; ::_thesis: verum end; then A6: |.p.| ^2 > 0 by SQUARE_1:12; percases ( (p `2) / |.p.| <= sn or (p `2) / |.p.| > sn ) ; supposeA7: (p `2) / |.p.| <= sn ; ::_thesis: (sn -FanMorphE) . x in K0 reconsider p9 = (sn -FanMorphE) . p as Point of (TOP-REAL 2) ; (sn -FanMorphE) . p = |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)))]| by A1, A3, A4, A7, Th84; then A8: p9 `1 = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) by EUCLID:52; A9: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; A10: 1 + sn > 0 by A1, XREAL_1:148; percases ( p `1 = 0 or p `1 <> 0 ) ; suppose p `1 = 0 ; ::_thesis: (sn -FanMorphE) . x in K0 hence (sn -FanMorphE) . x in K0 by A1, A2, Th82; ::_thesis: verum end; suppose p `1 <> 0 ; ::_thesis: (sn -FanMorphE) . x in K0 then 0 + ((p `2) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8; then ((p `2) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A9, XREAL_1:74; then ((p `2) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60; then ((p `2) / |.p.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (p `2) / |.p.| by SQUARE_1:52; then (- 1) - sn < ((p `2) / |.p.|) - sn by XREAL_1:9; then ((- 1) * (1 + sn)) / (1 + sn) < (((p `2) / |.p.|) - sn) / (1 + sn) by A10, XREAL_1:74; then A11: - 1 < (((p `2) / |.p.|) - sn) / (1 + sn) by A10, XCMPLX_1:89; ((p `2) / |.p.|) - sn <= 0 by A7, XREAL_1:47; then 1 ^2 > ((((p `2) / |.p.|) - sn) / (1 + sn)) ^2 by A10, A11, SQUARE_1:50; then 1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)) > 0 by SQUARE_1:25; then |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) > 0 by A5, XREAL_1:129; hence (sn -FanMorphE) . x in K0 by A1, A2, A8, JGRAPH_2:3; ::_thesis: verum end; end; end; supposeA12: (p `2) / |.p.| > sn ; ::_thesis: (sn -FanMorphE) . x in K0 reconsider p9 = (sn -FanMorphE) . p as Point of (TOP-REAL 2) ; (sn -FanMorphE) . p = |[(|.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))),(|.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)))]| by A1, A3, A4, A12, Th84; then A13: p9 `1 = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) by EUCLID:52; A14: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; A15: 1 - sn > 0 by A1, XREAL_1:149; percases ( p `1 = 0 or p `1 <> 0 ) ; suppose p `1 = 0 ; ::_thesis: (sn -FanMorphE) . x in K0 hence (sn -FanMorphE) . x in K0 by A1, A2, Th82; ::_thesis: verum end; suppose p `1 <> 0 ; ::_thesis: (sn -FanMorphE) . x in K0 then 0 + ((p `2) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8; then ((p `2) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A14, XREAL_1:74; then ((p `2) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60; then ((p `2) / |.p.|) ^2 < 1 by XCMPLX_1:76; then (p `2) / |.p.| < 1 by SQUARE_1:52; then ((p `2) / |.p.|) - sn < 1 - sn by XREAL_1:9; then (((p `2) / |.p.|) - sn) / (1 - sn) < (1 - sn) / (1 - sn) by A15, XREAL_1:74; then A16: (((p `2) / |.p.|) - sn) / (1 - sn) < 1 by A15, XCMPLX_1:60; ( - (1 - sn) < - 0 & ((p `2) / |.p.|) - sn >= sn - sn ) by A12, A15, XREAL_1:9, XREAL_1:24; then ((- 1) * (1 - sn)) / (1 - sn) < (((p `2) / |.p.|) - sn) / (1 - sn) by A15, XREAL_1:74; then - 1 < (((p `2) / |.p.|) - sn) / (1 - sn) by A15, XCMPLX_1:89; then 1 ^2 > ((((p `2) / |.p.|) - sn) / (1 - sn)) ^2 by A16, SQUARE_1:50; then 1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)) > 0 by SQUARE_1:25; then p9 `1 > 0 by A5, A13, XREAL_1:129; hence (sn -FanMorphE) . x in K0 by A1, A2, JGRAPH_2:3; ::_thesis: verum end; end; end; end; end; theorem Th99: :: JGRAPH_4:99 for sn being Real for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds (sn -FanMorphE) . x in K0 proof let sn be Real; ::_thesis: for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds (sn -FanMorphE) . x in K0 let x, K0 be set ; ::_thesis: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } implies (sn -FanMorphE) . x in K0 ) assume A1: ( - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: (sn -FanMorphE) . x in K0 then ex p being Point of (TOP-REAL 2) st ( p = x & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) ; hence (sn -FanMorphE) . x in K0 by A1, Th82; ::_thesis: verum end; theorem Th100: :: JGRAPH_4:100 for sn being Real for D being non empty Subset of (TOP-REAL 2) st - 1 < sn & sn < 1 & D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphE) | D & h is continuous ) proof ( |[0,1]| `1 = 0 & |[0,1]| `2 = 1 ) by EUCLID:52; then A1: |[0,1]| in { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } by JGRAPH_2:3; set Y1 = |[0,1]|; defpred S1[ Point of (TOP-REAL 2)] means $1 `1 >= 0 ; reconsider B0 = {(0. (TOP-REAL 2))} as Subset of (TOP-REAL 2) ; let sn be Real; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st - 1 < sn & sn < 1 & D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphE) | D & h is continuous ) let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 & D ` = {(0. (TOP-REAL 2))} implies ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphE) | D & h is continuous ) ) assume that A2: ( - 1 < sn & sn < 1 ) and A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphE) | D & h is continuous ) A4: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; A5: D = B0 ` by A3 .= NonZero (TOP-REAL 2) by SUBSET_1:def_4 ; { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) from JGRAPH_4:sch_1(A5); then reconsider K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A1; A6: K0 = the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; A7: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; A8: rng ((sn -FanMorphE) | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((sn -FanMorphE) | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) ) assume y in rng ((sn -FanMorphE) | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0) then consider x being set such that A9: x in dom ((sn -FanMorphE) | K0) and A10: y = ((sn -FanMorphE) | K0) . x by FUNCT_1:def_3; x in (dom (sn -FanMorphE)) /\ K0 by A9, RELAT_1:61; then A11: x in K0 by XBOOLE_0:def_4; K0 c= the carrier of (TOP-REAL 2) by A7, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A11; (sn -FanMorphE) . p = y by A10, A11, FUNCT_1:49; then y in K0 by A2, A11, Th98; hence y in the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; ::_thesis: verum end; A12: K0 c= the carrier of (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K0 or z in the carrier of (TOP-REAL 2) ) assume z in K0 ; ::_thesis: z in the carrier of (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & p8 `1 >= 0 & p8 <> 0. (TOP-REAL 2) ) ; hence z in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; ( |[0,1]| `1 = 0 & |[0,1]| `2 = 1 ) by EUCLID:52; then A13: |[0,1]| in { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } by JGRAPH_2:3; A14: the carrier of ((TOP-REAL 2) | D) = NonZero (TOP-REAL 2) by A5, PRE_TOPC:8; defpred S2[ Point of (TOP-REAL 2)] means $1 `1 <= 0 ; { p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) from JGRAPH_4:sch_1(A5); then reconsider K1 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A13; A15: ( K0 is closed & K1 is closed ) by A5, Th29, Th31; dom ((sn -FanMorphE) | K0) = (dom (sn -FanMorphE)) /\ K0 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K0 by FUNCT_2:def_1 .= K0 by A12, XBOOLE_1:28 ; then reconsider f = (sn -FanMorphE) | K0 as Function of (((TOP-REAL 2) | D) | K0),((TOP-REAL 2) | D) by A6, A8, FUNCT_2:2, XBOOLE_1:1; A16: K1 = the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; A17: rng ((sn -FanMorphE) | K1) c= the carrier of (((TOP-REAL 2) | D) | K1) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((sn -FanMorphE) | K1) or y in the carrier of (((TOP-REAL 2) | D) | K1) ) assume y in rng ((sn -FanMorphE) | K1) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K1) then consider x being set such that A18: x in dom ((sn -FanMorphE) | K1) and A19: y = ((sn -FanMorphE) | K1) . x by FUNCT_1:def_3; x in (dom (sn -FanMorphE)) /\ K1 by A18, RELAT_1:61; then A20: x in K1 by XBOOLE_0:def_4; K1 c= the carrier of (TOP-REAL 2) by A7, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A20; (sn -FanMorphE) . p = y by A19, A20, FUNCT_1:49; then y in K1 by A2, A20, Th99; hence y in the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; ::_thesis: verum end; A21: K1 c= the carrier of (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in the carrier of (TOP-REAL 2) ) assume z in K1 ; ::_thesis: z in the carrier of (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & p8 `1 <= 0 & p8 <> 0. (TOP-REAL 2) ) ; hence z in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; dom ((sn -FanMorphE) | K1) = (dom (sn -FanMorphE)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by A21, XBOOLE_1:28 ; then reconsider g = (sn -FanMorphE) | K1 as Function of (((TOP-REAL 2) | D) | K1),((TOP-REAL 2) | D) by A16, A17, FUNCT_2:2, XBOOLE_1:1; A22: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5; A23: D c= K0 \/ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in K0 \/ K1 ) assume A24: x in D ; ::_thesis: x in K0 \/ K1 then reconsider px = x as Point of (TOP-REAL 2) ; not x in {(0. (TOP-REAL 2))} by A5, A24, XBOOLE_0:def_5; then ( ( px `1 >= 0 & px <> 0. (TOP-REAL 2) ) or ( px `1 <= 0 & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1; then ( x in K0 or x in K1 ) ; hence x in K0 \/ K1 by XBOOLE_0:def_3; ::_thesis: verum end; A25: dom f = K0 by A6, FUNCT_2:def_1; A26: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5; A27: for x being set st x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) holds f . x = g . x proof let x be set ; ::_thesis: ( x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) implies f . x = g . x ) assume A28: x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) ; ::_thesis: f . x = g . x then x in K0 by A26, XBOOLE_0:def_4; then f . x = (sn -FanMorphE) . x by FUNCT_1:49; hence f . x = g . x by A22, A28, FUNCT_1:49; ::_thesis: verum end; D = [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; then A29: ([#] (((TOP-REAL 2) | D) | K0)) \/ ([#] (((TOP-REAL 2) | D) | K1)) = [#] ((TOP-REAL 2) | D) by A26, A22, A23, XBOOLE_0:def_10; A30: ( f is continuous & g is continuous ) by A2, A5, Th95, Th96; then consider h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) such that A31: h = f +* g and h is continuous by A26, A22, A29, A15, A27, JGRAPH_2:1; A32: dom h = the carrier of ((TOP-REAL 2) | D) by FUNCT_2:def_1; A33: dom g = K1 by A16, FUNCT_2:def_1; ( K0 = [#] (((TOP-REAL 2) | D) | K0) & K1 = [#] (((TOP-REAL 2) | D) | K1) ) by PRE_TOPC:def_5; then A34: f tolerates g by A27, A25, A33, PARTFUN1:def_4; A35: for x being set st x in dom h holds h . x = ((sn -FanMorphE) | D) . x proof let x be set ; ::_thesis: ( x in dom h implies h . x = ((sn -FanMorphE) | D) . x ) assume A36: x in dom h ; ::_thesis: h . x = ((sn -FanMorphE) | D) . x then reconsider p = x as Point of (TOP-REAL 2) by A14, XBOOLE_0:def_5; not x in {(0. (TOP-REAL 2))} by A14, A36, XBOOLE_0:def_5; then A37: x <> 0. (TOP-REAL 2) by TARSKI:def_1; A38: x in (D `) ` by A32, A36, PRE_TOPC:8; now__::_thesis:_(_(_x_in_K0_&_h_._x_=_((sn_-FanMorphE)_|_D)_._x_)_or_(_not_x_in_K0_&_h_._x_=_((sn_-FanMorphE)_|_D)_._x_)_) percases ( x in K0 or not x in K0 ) ; caseA39: x in K0 ; ::_thesis: h . x = ((sn -FanMorphE) | D) . x A40: ((sn -FanMorphE) | D) . p = (sn -FanMorphE) . p by A38, FUNCT_1:49 .= f . p by A39, FUNCT_1:49 ; h . p = (g +* f) . p by A31, A34, FUNCT_4:34 .= f . p by A25, A39, FUNCT_4:13 ; hence h . x = ((sn -FanMorphE) | D) . x by A40; ::_thesis: verum end; case not x in K0 ; ::_thesis: h . x = ((sn -FanMorphE) | D) . x then not p `1 >= 0 by A37; then A41: x in K1 by A37; ((sn -FanMorphE) | D) . p = (sn -FanMorphE) . p by A38, FUNCT_1:49 .= g . p by A41, FUNCT_1:49 ; hence h . x = ((sn -FanMorphE) | D) . x by A31, A33, A41, FUNCT_4:13; ::_thesis: verum end; end; end; hence h . x = ((sn -FanMorphE) | D) . x ; ::_thesis: verum end; dom (sn -FanMorphE) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then dom ((sn -FanMorphE) | D) = the carrier of (TOP-REAL 2) /\ D by RELAT_1:61 .= the carrier of ((TOP-REAL 2) | D) by A4, XBOOLE_1:28 ; then f +* g = (sn -FanMorphE) | D by A31, A32, A35, FUNCT_1:2; hence ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphE) | D & h is continuous ) by A26, A22, A29, A30, A15, A27, JGRAPH_2:1; ::_thesis: verum end; theorem Th101: :: JGRAPH_4:101 for sn being Real st - 1 < sn & sn < 1 holds ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphE & h is continuous ) proof reconsider D = NonZero (TOP-REAL 2) as non empty Subset of (TOP-REAL 2) by JGRAPH_2:9; let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphE & h is continuous ) ) assume that A1: - 1 < sn and A2: sn < 1 ; ::_thesis: ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphE & h is continuous ) reconsider f = sn -FanMorphE as Function of (TOP-REAL 2),(TOP-REAL 2) ; A3: f . (0. (TOP-REAL 2)) = 0. (TOP-REAL 2) by Th82, JGRAPH_2:3; A4: for p being Point of ((TOP-REAL 2) | D) holds f . p <> f . (0. (TOP-REAL 2)) proof let p be Point of ((TOP-REAL 2) | D); ::_thesis: f . p <> f . (0. (TOP-REAL 2)) A5: [#] ((TOP-REAL 2) | D) = D by PRE_TOPC:def_5; then reconsider q = p as Point of (TOP-REAL 2) by XBOOLE_0:def_5; not p in {(0. (TOP-REAL 2))} by A5, XBOOLE_0:def_5; then A6: not p = 0. (TOP-REAL 2) by TARSKI:def_1; now__::_thesis:_(_(_(q_`2)_/_|.q.|_>=_sn_&_q_`1_>=_0_&_f_._p_<>_f_._(0._(TOP-REAL_2))_)_or_(_(q_`2)_/_|.q.|_<_sn_&_q_`1_>=_0_&_f_._p_<>_f_._(0._(TOP-REAL_2))_)_or_(_q_`1_<_0_&_f_._p_<>_f_._(0._(TOP-REAL_2))_)_) percases ( ( (q `2) / |.q.| >= sn & q `1 >= 0 ) or ( (q `2) / |.q.| < sn & q `1 >= 0 ) or q `1 < 0 ) ; caseA7: ( (q `2) / |.q.| >= sn & q `1 >= 0 ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) set q9 = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]|; A8: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)) by EUCLID:52; now__::_thesis:_not_|[(|.q.|_*_(sqrt_(1_-_(((((q_`2)_/_|.q.|)_-_sn)_/_(1_-_sn))_^2)))),(|.q.|_*_((((q_`2)_/_|.q.|)_-_sn)_/_(1_-_sn)))]|_=_0._(TOP-REAL_2) assume A9: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction A10: |.q.| <> 0 by A6, TOPRNS_1:24; then sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) = sqrt (1 - (0 ^2)) by A8, A9, JGRAPH_2:3, XCMPLX_1:6 .= 1 by SQUARE_1:18 ; hence contradiction by A9, A10, EUCLID:52, JGRAPH_2:3; ::_thesis: verum end; hence f . p <> f . (0. (TOP-REAL 2)) by A1, A2, A3, A6, A7, Th84; ::_thesis: verum end; caseA11: ( (q `2) / |.q.| < sn & q `1 >= 0 ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) set q9 = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|; A12: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) by EUCLID:52; now__::_thesis:_not_|[(|.q.|_*_(sqrt_(1_-_(((((q_`2)_/_|.q.|)_-_sn)_/_(1_+_sn))_^2)))),(|.q.|_*_((((q_`2)_/_|.q.|)_-_sn)_/_(1_+_sn)))]|_=_0._(TOP-REAL_2) assume A13: |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction A14: |.q.| <> 0 by A6, TOPRNS_1:24; then sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) = sqrt (1 - (0 ^2)) by A12, A13, JGRAPH_2:3, XCMPLX_1:6 .= 1 by SQUARE_1:18 ; hence contradiction by A13, A14, EUCLID:52, JGRAPH_2:3; ::_thesis: verum end; hence f . p <> f . (0. (TOP-REAL 2)) by A1, A2, A3, A6, A11, Th84; ::_thesis: verum end; case q `1 < 0 ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) then f . p = p by Th82; hence f . p <> f . (0. (TOP-REAL 2)) by A6, Th82, JGRAPH_2:3; ::_thesis: verum end; end; end; hence f . p <> f . (0. (TOP-REAL 2)) ; ::_thesis: verum end; A15: for V being Subset of (TOP-REAL 2) st f . (0. (TOP-REAL 2)) in V & V is open holds ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) proof reconsider u0 = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:67; let V be Subset of (TOP-REAL 2); ::_thesis: ( f . (0. (TOP-REAL 2)) in V & V is open implies ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) ) reconsider VV = V as Subset of (TopSpaceMetr (Euclid 2)) by Lm11; assume that A16: f . (0. (TOP-REAL 2)) in V and A17: V is open ; ::_thesis: ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) VV is open by A17, Lm11, PRE_TOPC:30; then consider r being real number such that A18: r > 0 and A19: Ball (u0,r) c= V by A3, A16, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider W1 = Ball (u0,r) as Subset of (TOP-REAL 2) ; A20: W1 is open by GOBOARD6:3; A21: f .: W1 c= W1 proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f .: W1 or z in W1 ) assume z in f .: W1 ; ::_thesis: z in W1 then consider y being set such that A22: y in dom f and A23: y in W1 and A24: z = f . y by FUNCT_1:def_6; z in rng f by A22, A24, FUNCT_1:def_3; then reconsider qz = z as Point of (TOP-REAL 2) ; reconsider q = y as Point of (TOP-REAL 2) by A22; reconsider qy = q as Point of (Euclid 2) by EUCLID:67; reconsider pz = qz as Point of (Euclid 2) by EUCLID:67; dist (u0,qy) < r by A23, METRIC_1:11; then A25: |.((0. (TOP-REAL 2)) - q).| < r by JGRAPH_1:28; now__::_thesis:_(_(_q_`1_<=_0_&_z_in_W1_)_or_(_q_<>_0._(TOP-REAL_2)_&_(q_`2)_/_|.q.|_>=_sn_&_q_`1_>=_0_&_z_in_W1_)_or_(_q_<>_0._(TOP-REAL_2)_&_(q_`2)_/_|.q.|_<_sn_&_q_`1_>=_0_&_z_in_W1_)_) percases ( q `1 <= 0 or ( q <> 0. (TOP-REAL 2) & (q `2) / |.q.| >= sn & q `1 >= 0 ) or ( q <> 0. (TOP-REAL 2) & (q `2) / |.q.| < sn & q `1 >= 0 ) ) by JGRAPH_2:3; case q `1 <= 0 ; ::_thesis: z in W1 hence z in W1 by A23, A24, Th82; ::_thesis: verum end; caseA26: ( q <> 0. (TOP-REAL 2) & (q `2) / |.q.| >= sn & q `1 >= 0 ) ; ::_thesis: z in W1 then A27: ((q `2) / |.q.|) - sn >= 0 by XREAL_1:48; 0 <= (q `1) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A28: ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; A29: 1 - sn > 0 by A2, XREAL_1:149; |.q.| <> 0 by A26, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A28, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `2) / |.q.| by SQUARE_1:51; then 1 - sn >= ((q `2) / |.q.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((q `2) / |.q.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A29, XREAL_1:72; then - 1 <= (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A29, XCMPLX_1:197; then ((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A29, A27, SQUARE_1:49; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A30: 1 - ((- ((((q `2) / |.q.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; A31: (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, A26, Th84; then A32: qz `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)) by A24, EUCLID:52; qz `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) by A24, A31, EUCLID:52; then A33: (qz `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) by A30, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A32, A33 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; then A34: |.qz.| = |.q.| by SQUARE_1:22; |.(- q).| < r by A25, EUCLID:27; then |.q.| < r by TOPRNS_1:26; then |.(- qz).| < r by A34, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - qz).| < r by EUCLID:27; then dist (u0,pz) < r by JGRAPH_1:28; hence z in W1 by METRIC_1:11; ::_thesis: verum end; caseA35: ( q <> 0. (TOP-REAL 2) & (q `2) / |.q.| < sn & q `1 >= 0 ) ; ::_thesis: z in W1 0 <= (q `1) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A36: ((q `2) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; A37: 1 + sn > 0 by A1, XREAL_1:148; |.q.| <> 0 by A35, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `2) ^2) / (|.q.| ^2) <= 1 by A36, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `2) / |.q.| by SQUARE_1:51; then - (- 1) >= - ((q `2) / |.q.|) by XREAL_1:24; then 1 + sn >= (- ((q `2) / |.q.|)) + sn by XREAL_1:7; then A38: (- (((q `2) / |.q.|) - sn)) / (1 + sn) <= 1 by A37, XREAL_1:185; sn - ((q `2) / |.q.|) >= 0 by A35, XREAL_1:48; then - 1 <= (- (((q `2) / |.q.|) - sn)) / (1 + sn) by A37; then ((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A38, SQUARE_1:49; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then A39: 1 - ((- ((((q `2) / |.q.|) - sn) / (1 + sn))) ^2) >= 0 by XCMPLX_1:187; A40: (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A1, A2, A35, Th84; then A41: qz `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) by A24, EUCLID:52; qz `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) by A24, A40, EUCLID:52; then A42: (qz `1) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) by A39, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A41, A42 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; then A43: |.qz.| = |.q.| by SQUARE_1:22; |.(- q).| < r by A25, EUCLID:27; then |.q.| < r by TOPRNS_1:26; then |.(- qz).| < r by A43, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - qz).| < r by EUCLID:27; then dist (u0,pz) < r by JGRAPH_1:28; hence z in W1 by METRIC_1:11; ::_thesis: verum end; end; end; hence z in W1 ; ::_thesis: verum end; u0 in W1 by A18, GOBOARD6:1; hence ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) by A19, A20, A21, XBOOLE_1:1; ::_thesis: verum end; A44: D ` = {(0. (TOP-REAL 2))} by JGRAPH_3:20; then ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (sn -FanMorphE) | D & h is continuous ) by A1, A2, Th100; hence ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphE & h is continuous ) by A3, A44, A4, A15, JGRAPH_3:3; ::_thesis: verum end; theorem Th102: :: JGRAPH_4:102 for sn being Real st - 1 < sn & sn < 1 holds sn -FanMorphE is one-to-one proof let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies sn -FanMorphE is one-to-one ) assume that A1: - 1 < sn and A2: sn < 1 ; ::_thesis: sn -FanMorphE is one-to-one for x1, x2 being set st x1 in dom (sn -FanMorphE) & x2 in dom (sn -FanMorphE) & (sn -FanMorphE) . x1 = (sn -FanMorphE) . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom (sn -FanMorphE) & x2 in dom (sn -FanMorphE) & (sn -FanMorphE) . x1 = (sn -FanMorphE) . x2 implies x1 = x2 ) assume that A3: x1 in dom (sn -FanMorphE) and A4: x2 in dom (sn -FanMorphE) and A5: (sn -FanMorphE) . x1 = (sn -FanMorphE) . x2 ; ::_thesis: x1 = x2 reconsider p2 = x2 as Point of (TOP-REAL 2) by A4; reconsider p1 = x1 as Point of (TOP-REAL 2) by A3; set q = p1; set p = p2; A6: 1 - sn > 0 by A2, XREAL_1:149; now__::_thesis:_(_(_p1_`1_<=_0_&_x1_=_x2_)_or_(_(p1_`2)_/_|.p1.|_>=_sn_&_p1_`1_>=_0_&_p1_<>_0._(TOP-REAL_2)_&_x1_=_x2_)_or_(_(p1_`2)_/_|.p1.|_<_sn_&_p1_`1_>=_0_&_p1_<>_0._(TOP-REAL_2)_&_x1_=_x2_)_) percases ( p1 `1 <= 0 or ( (p1 `2) / |.p1.| >= sn & p1 `1 >= 0 & p1 <> 0. (TOP-REAL 2) ) or ( (p1 `2) / |.p1.| < sn & p1 `1 >= 0 & p1 <> 0. (TOP-REAL 2) ) ) by JGRAPH_2:3; caseA7: p1 `1 <= 0 ; ::_thesis: x1 = x2 then A8: (sn -FanMorphE) . p1 = p1 by Th82; now__::_thesis:_(_(_p2_`1_<=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_>=_sn_&_p2_`1_>=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_<_sn_&_p2_`1_>=_0_&_x1_=_x2_)_) percases ( p2 `1 <= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 >= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 >= 0 ) ) by JGRAPH_2:3; case p2 `1 <= 0 ; ::_thesis: x1 = x2 hence x1 = x2 by A5, A8, Th82; ::_thesis: verum end; caseA9: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 >= 0 ) ; ::_thesis: x1 = x2 then A10: ((p2 `2) / |.p2.|) - sn >= 0 by XREAL_1:48; set p4 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|; A11: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; A12: |.p2.| <> 0 by A9, TOPRNS_1:24; then A13: |.p2.| ^2 > 0 by SQUARE_1:12; 0 <= (p2 `1) ^2 by XREAL_1:63; then 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) by XREAL_1:7; then ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by A11, XREAL_1:72; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A13, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p2 `2) / |.p2.| by SQUARE_1:51; then 1 - sn >= ((p2 `2) / |.p2.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((p2 `2) / |.p2.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((p2 `2) / |.p2.|) - sn)) / (1 - sn) by A6, XREAL_1:72; then A14: - 1 <= (- (((p2 `2) / |.p2.|) - sn)) / (1 - sn) by A6, XCMPLX_1:197; A15: (sn -FanMorphE) . p2 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| by A1, A2, A9, Th84; ((p2 `2) / |.p2.|) - sn >= 0 by A9, XREAL_1:48; then ((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A6, A14, SQUARE_1:49; then A16: 1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p2 `2) / |.p2.|) - sn)) ^2) / ((1 - sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p2 `2) / |.p2.|) - sn) ^2) / ((1 - sn) ^2))) >= 0 ; then sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)) >= 0 by XCMPLX_1:76; then ( |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `1 = |.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))) & p1 `1 = 0 ) by A5, A7, A8, A15, EUCLID:52; then A17: sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)) = 0 by A5, A8, A15, A12, XCMPLX_1:6; 1 - ((- ((((p2 `2) / |.p2.|) - sn) / (1 - sn))) ^2) >= 0 by A16, XCMPLX_1:187; then 1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2) = 0 by A17, SQUARE_1:24; then 1 = (((p2 `2) / |.p2.|) - sn) / (1 - sn) by A6, A10, SQUARE_1:18, SQUARE_1:22; then 1 * (1 - sn) = ((p2 `2) / |.p2.|) - sn by A6, XCMPLX_1:87; then 1 * |.p2.| = p2 `2 by A9, TOPRNS_1:24, XCMPLX_1:87; then p2 `1 = 0 by A11, XCMPLX_1:6; hence x1 = x2 by A5, A8, Th82; ::_thesis: verum end; caseA18: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 >= 0 ) ; ::_thesis: x1 = x2 then A19: |.p2.| <> 0 by TOPRNS_1:24; then A20: |.p2.| ^2 > 0 by SQUARE_1:12; set p4 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|; A21: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; A22: 1 + sn > 0 by A1, XREAL_1:148; A23: ((p2 `2) / |.p2.|) - sn <= 0 by A18, XREAL_1:47; then A24: - 1 <= (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) by A22; A25: (sn -FanMorphE) . p2 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| by A1, A2, A18, Th84; 0 <= (p2 `1) ^2 by XREAL_1:63; then 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) by XREAL_1:7; then ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by A21, XREAL_1:72; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A20, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (- ((p2 `2) / |.p2.|)) ^2 <= 1 ; then 1 >= - ((p2 `2) / |.p2.|) by SQUARE_1:51; then 1 + sn >= (- ((p2 `2) / |.p2.|)) + sn by XREAL_1:7; then (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) <= 1 by A22, XREAL_1:185; then ((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A24, SQUARE_1:49; then A26: 1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p2 `2) / |.p2.|) - sn)) ^2) / ((1 + sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p2 `2) / |.p2.|) - sn) ^2) / ((1 + sn) ^2))) >= 0 ; then sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)) >= 0 by XCMPLX_1:76; then ( |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `1 = |.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))) & p1 `1 = 0 ) by A5, A7, A8, A25, EUCLID:52; then A27: sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)) = 0 by A5, A8, A25, A19, XCMPLX_1:6; 1 - ((- ((((p2 `2) / |.p2.|) - sn) / (1 + sn))) ^2) >= 0 by A26, XCMPLX_1:187; then 1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2) = 0 by A27, SQUARE_1:24; then 1 = sqrt ((- ((((p2 `2) / |.p2.|) - sn) / (1 + sn))) ^2) by SQUARE_1:18; then 1 = - ((((p2 `2) / |.p2.|) - sn) / (1 + sn)) by A22, A23, SQUARE_1:22; then 1 = (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) by XCMPLX_1:187; then 1 * (1 + sn) = - (((p2 `2) / |.p2.|) - sn) by A22, XCMPLX_1:87; then (1 + sn) - sn = - ((p2 `2) / |.p2.|) ; then 1 = (- (p2 `2)) / |.p2.| by XCMPLX_1:187; then 1 * |.p2.| = - (p2 `2) by A18, TOPRNS_1:24, XCMPLX_1:87; then ((p2 `2) ^2) - ((p2 `2) ^2) = (p2 `1) ^2 by A21, XCMPLX_1:26; then p2 `1 = 0 by XCMPLX_1:6; hence x1 = x2 by A5, A8, Th82; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; caseA28: ( (p1 `2) / |.p1.| >= sn & p1 `1 >= 0 & p1 <> 0. (TOP-REAL 2) ) ; ::_thesis: x1 = x2 then |.p1.| <> 0 by TOPRNS_1:24; then A29: |.p1.| ^2 > 0 by SQUARE_1:12; set q4 = |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]|; A30: |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `2 = |.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)) by EUCLID:52; A31: (sn -FanMorphE) . p1 = |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| by A1, A2, A28, Th84; A32: |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `1 = |.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))) by EUCLID:52; now__::_thesis:_(_(_p2_`1_<=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_>=_sn_&_p2_`1_>=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_<_sn_&_p2_`1_>=_0_&_x1_=_x2_)_) percases ( p2 `1 <= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 >= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 >= 0 ) ) by JGRAPH_2:3; caseA33: p2 `1 <= 0 ; ::_thesis: x1 = x2 then A34: (sn -FanMorphE) . p2 = p2 by Th82; A35: |.p1.| <> 0 by A28, TOPRNS_1:24; then A36: |.p1.| ^2 > 0 by SQUARE_1:12; A37: ((p1 `2) / |.p1.|) - sn >= 0 by A28, XREAL_1:48; A38: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1; A39: ((p1 `2) / |.p1.|) - sn >= 0 by A28, XREAL_1:48; A40: 1 - sn > 0 by A2, XREAL_1:149; 0 <= (p1 `1) ^2 by XREAL_1:63; then 0 + ((p1 `2) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) by XREAL_1:7; then ((p1 `2) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by A38, XREAL_1:72; then ((p1 `2) ^2) / (|.p1.| ^2) <= 1 by A36, XCMPLX_1:60; then ((p1 `2) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p1 `2) / |.p1.| by SQUARE_1:51; then 1 - sn >= ((p1 `2) / |.p1.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((p1 `2) / |.p1.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((p1 `2) / |.p1.|) - sn)) / (1 - sn) by A40, XREAL_1:72; then - 1 <= (- (((p1 `2) / |.p1.|) - sn)) / (1 - sn) by A40, XCMPLX_1:197; then ((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A40, A37, SQUARE_1:49; then A41: 1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p1 `2) / |.p1.|) - sn)) ^2) / ((1 - sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) ^2) / ((1 - sn) ^2))) >= 0 ; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)) >= 0 by XCMPLX_1:76; then p2 `1 = 0 by A5, A31, A33, A34, EUCLID:52; then A42: sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)) = 0 by A5, A31, A32, A34, A35, XCMPLX_1:6; 1 - ((- ((((p1 `2) / |.p1.|) - sn) / (1 - sn))) ^2) >= 0 by A41, XCMPLX_1:187; then 1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2) = 0 by A42, SQUARE_1:24; then 1 = (((p1 `2) / |.p1.|) - sn) / (1 - sn) by A40, A39, SQUARE_1:18, SQUARE_1:22; then 1 * (1 - sn) = ((p1 `2) / |.p1.|) - sn by A40, XCMPLX_1:87; then 1 * |.p1.| = p1 `2 by A28, TOPRNS_1:24, XCMPLX_1:87; then p1 `1 = 0 by A38, XCMPLX_1:6; hence x1 = x2 by A5, A34, Th82; ::_thesis: verum end; caseA43: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 >= 0 ) ; ::_thesis: x1 = x2 0 <= (p1 `1) ^2 by XREAL_1:63; then ( |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) & 0 + ((p1 `2) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p1 `2) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by XREAL_1:72; then ((p1 `2) ^2) / (|.p1.| ^2) <= 1 by A29, XCMPLX_1:60; then ((p1 `2) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p1 `2) / |.p1.| by SQUARE_1:51; then 1 - sn >= ((p1 `2) / |.p1.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((p1 `2) / |.p1.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((p1 `2) / |.p1.|) - sn)) / (1 - sn) by A6, XREAL_1:72; then A44: - 1 <= (- (((p1 `2) / |.p1.|) - sn)) / (1 - sn) by A6, XCMPLX_1:197; ((p1 `2) / |.p1.|) - sn >= 0 by A28, XREAL_1:48; then ((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A6, A44, SQUARE_1:49; then 1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A45: 1 - ((- ((((p1 `2) / |.p1.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `1 = |.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))) by EUCLID:52; then A46: (|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `1) ^2 = (|.p1.| ^2) * ((sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2))) ^2) .= (|.p1.| ^2) * (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)) by A45, SQUARE_1:def_2 ; A47: |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `2 = |.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)) by EUCLID:52; |.|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]|.| ^2 = ((|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `1) ^2) + ((|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]| `2) ^2) by JGRAPH_3:1 .= |.p1.| ^2 by A47, A46 ; then A48: sqrt (|.|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]|.| ^2) = |.p1.| by SQUARE_1:22; then A49: |.|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 - sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn)))]|.| = |.p1.| by SQUARE_1:22; 0 <= (p2 `1) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A50: ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; |.p2.| <> 0 by A43, TOPRNS_1:24; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A50, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p2 `2) / |.p2.| by SQUARE_1:51; then 1 - sn >= ((p2 `2) / |.p2.|) - sn by XREAL_1:9; then - (1 - sn) <= - (((p2 `2) / |.p2.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) <= (- (((p2 `2) / |.p2.|) - sn)) / (1 - sn) by A6, XREAL_1:72; then A51: - 1 <= (- (((p2 `2) / |.p2.|) - sn)) / (1 - sn) by A6, XCMPLX_1:197; ((p2 `2) / |.p2.|) - sn >= 0 by A43, XREAL_1:48; then ((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2 <= 1 ^2 by A6, A51, SQUARE_1:49; then 1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 - sn)) ^2) >= 0 by XREAL_1:48; then A52: 1 - ((- ((((p2 `2) / |.p2.|) - sn) / (1 - sn))) ^2) >= 0 by XCMPLX_1:187; set p4 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|; A53: |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)) by EUCLID:52; |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `1 = |.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))) by EUCLID:52; then A54: (|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `1) ^2 = (|.p2.| ^2) * ((sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2))) ^2) .= (|.p2.| ^2) * (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)) by A52, SQUARE_1:def_2 ; |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|.| ^2 = ((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `1) ^2) + ((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `2) ^2) by JGRAPH_3:1 .= |.p2.| ^2 by A53, A54 ; then A55: sqrt (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|.| ^2) = |.p2.| by SQUARE_1:22; then A56: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|.| = |.p2.| by SQUARE_1:22; A57: (sn -FanMorphE) . p2 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| by A1, A2, A43, Th84; then (((p2 `2) / |.p2.|) - sn) / (1 - sn) = (|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 - sn))) / |.p2.| by A5, A31, A30, A43, A53, TOPRNS_1:24, XCMPLX_1:89; then (((p2 `2) / |.p2.|) - sn) / (1 - sn) = (((p1 `2) / |.p1.|) - sn) / (1 - sn) by A5, A31, A43, A57, A48, A55, TOPRNS_1:24, XCMPLX_1:89; then ((((p2 `2) / |.p2.|) - sn) / (1 - sn)) * (1 - sn) = ((p1 `2) / |.p1.|) - sn by A6, XCMPLX_1:87; then ((p2 `2) / |.p2.|) - sn = ((p1 `2) / |.p1.|) - sn by A6, XCMPLX_1:87; then ((p2 `2) / |.p2.|) * |.p2.| = p1 `2 by A5, A31, A43, A57, A49, A56, TOPRNS_1:24, XCMPLX_1:87; then A58: p2 `2 = p1 `2 by A43, TOPRNS_1:24, XCMPLX_1:87; A59: p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53; ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1; then p2 `1 = sqrt ((p1 `1) ^2) by A5, A31, A43, A57, A49, A56, A58, SQUARE_1:22; then p2 `1 = p1 `1 by A28, SQUARE_1:22; hence x1 = x2 by A58, A59, EUCLID:53; ::_thesis: verum end; caseA60: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 >= 0 ) ; ::_thesis: x1 = x2 then ((p2 `2) / |.p2.|) - sn < 0 by XREAL_1:49; then A61: (((p2 `2) / |.p2.|) - sn) / (1 + sn) < 0 by A1, XREAL_1:141, XREAL_1:148; set p4 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|; A62: ( |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)) & ((p1 `2) / |.p1.|) - sn >= 0 ) by A28, EUCLID:52, XREAL_1:48; A63: 1 - sn > 0 by A2, XREAL_1:149; ( (sn -FanMorphE) . p2 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| & |.p2.| <> 0 ) by A1, A2, A60, Th84, TOPRNS_1:24; hence x1 = x2 by A5, A31, A30, A61, A62, A63, XREAL_1:132; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; caseA64: ( (p1 `2) / |.p1.| < sn & p1 `1 >= 0 & p1 <> 0. (TOP-REAL 2) ) ; ::_thesis: x1 = x2 then A65: |.p1.| <> 0 by TOPRNS_1:24; then A66: |.p1.| ^2 > 0 by SQUARE_1:12; set q4 = |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]|; A67: |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `2 = |.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)) by EUCLID:52; A68: (sn -FanMorphE) . p1 = |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| by A1, A2, A64, Th84; A69: |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `1 = |.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))) by EUCLID:52; now__::_thesis:_(_(_p2_`1_<=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_>=_sn_&_p2_`1_>=_0_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(p2_`2)_/_|.p2.|_<_sn_&_p2_`1_>=_0_&_x1_=_x2_)_) percases ( p2 `1 <= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 >= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 >= 0 ) ) by JGRAPH_2:3; caseA70: p2 `1 <= 0 ; ::_thesis: x1 = x2 A71: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1; A72: 1 + sn > 0 by A1, XREAL_1:148; 0 <= (p1 `1) ^2 by XREAL_1:63; then 0 + ((p1 `2) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) by XREAL_1:7; then ((p1 `2) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by A71, XREAL_1:72; then ((p1 `2) ^2) / (|.p1.| ^2) <= 1 by A66, XCMPLX_1:60; then ((p1 `2) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then (- ((p1 `2) / |.p1.|)) ^2 <= 1 ; then 1 >= - ((p1 `2) / |.p1.|) by SQUARE_1:51; then 1 + sn >= (- ((p1 `2) / |.p1.|)) + sn by XREAL_1:7; then A73: (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) <= 1 by A72, XREAL_1:185; A74: ((p1 `2) / |.p1.|) - sn <= 0 by A64, XREAL_1:47; then - 1 <= (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) by A72; then ((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A73, SQUARE_1:49; then A75: 1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then A76: 1 - ((- ((((p1 `2) / |.p1.|) - sn) / (1 + sn))) ^2) >= 0 by XCMPLX_1:187; A77: (sn -FanMorphE) . p2 = p2 by A70, Th82; sqrt (1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2)) >= 0 by A75, SQUARE_1:def_2; then sqrt (1 - (((- (((p1 `2) / |.p1.|) - sn)) ^2) / ((1 + sn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) ^2) / ((1 + sn) ^2))) >= 0 ; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)) >= 0 by XCMPLX_1:76; then p2 `1 = 0 by A5, A68, A70, A77, EUCLID:52; then sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)) = 0 by A5, A68, A69, A65, A77, XCMPLX_1:6; then 1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2) = 0 by A76, SQUARE_1:24; then 1 = sqrt ((- ((((p1 `2) / |.p1.|) - sn) / (1 + sn))) ^2) by SQUARE_1:18; then 1 = - ((((p1 `2) / |.p1.|) - sn) / (1 + sn)) by A72, A74, SQUARE_1:22; then 1 = (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) by XCMPLX_1:187; then 1 * (1 + sn) = - (((p1 `2) / |.p1.|) - sn) by A72, XCMPLX_1:87; then (1 + sn) - sn = - ((p1 `2) / |.p1.|) ; then 1 = (- (p1 `2)) / |.p1.| by XCMPLX_1:187; then 1 * |.p1.| = - (p1 `2) by A64, TOPRNS_1:24, XCMPLX_1:87; then ((p1 `2) ^2) - ((p1 `2) ^2) = (p1 `1) ^2 by A71, XCMPLX_1:26; then p1 `1 = 0 by XCMPLX_1:6; hence x1 = x2 by A5, A77, Th82; ::_thesis: verum end; caseA78: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| >= sn & p2 `1 >= 0 ) ; ::_thesis: x1 = x2 set p4 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]|; A79: ( |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)) & |.p1.| <> 0 ) by A64, EUCLID:52, TOPRNS_1:24; ((p1 `2) / |.p1.|) - sn < 0 by A64, XREAL_1:49; then A80: (((p1 `2) / |.p1.|) - sn) / (1 + sn) < 0 by A1, XREAL_1:141, XREAL_1:148; A81: 1 - sn > 0 by A2, XREAL_1:149; ( (sn -FanMorphE) . p2 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 - sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 - sn)))]| & ((p2 `2) / |.p2.|) - sn >= 0 ) by A1, A2, A78, Th84, XREAL_1:48; hence x1 = x2 by A5, A68, A67, A80, A79, A81, XREAL_1:132; ::_thesis: verum end; caseA82: ( p2 <> 0. (TOP-REAL 2) & (p2 `2) / |.p2.| < sn & p2 `1 >= 0 ) ; ::_thesis: x1 = x2 0 <= (p2 `1) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A83: ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; A84: 1 + sn > 0 by A1, XREAL_1:148; 0 <= (p1 `1) ^2 by XREAL_1:63; then ( |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) & 0 + ((p1 `2) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p1 `2) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by XREAL_1:72; then ((p1 `2) ^2) / (|.p1.| ^2) <= 1 by A66, XCMPLX_1:60; then ((p1 `2) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p1 `2) / |.p1.| by SQUARE_1:51; then (- 1) - sn <= ((p1 `2) / |.p1.|) - sn by XREAL_1:9; then - ((- 1) - sn) >= - (((p1 `2) / |.p1.|) - sn) by XREAL_1:24; then A85: (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) <= 1 by A84, XREAL_1:185; ((p1 `2) / |.p1.|) - sn <= 0 by A64, XREAL_1:47; then - 1 <= (- (((p1 `2) / |.p1.|) - sn)) / (1 + sn) by A84; then ((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A85, SQUARE_1:49; then 1 - (((- (((p1 `2) / |.p1.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then A86: 1 - ((- ((((p1 `2) / |.p1.|) - sn) / (1 + sn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `1 = |.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))) by EUCLID:52; then A87: (|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `1) ^2 = (|.p1.| ^2) * ((sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2))) ^2) .= (|.p1.| ^2) * (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)) by A86, SQUARE_1:def_2 ; A88: |[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `2 = |.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)) by EUCLID:52; set p4 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|; A89: |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)) by EUCLID:52; |.p2.| <> 0 by A82, TOPRNS_1:24; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A83, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p2 `2) / |.p2.| by SQUARE_1:51; then (- 1) - sn <= ((p2 `2) / |.p2.|) - sn by XREAL_1:9; then - ((- 1) - sn) >= - (((p2 `2) / |.p2.|) - sn) by XREAL_1:24; then A90: (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) <= 1 by A84, XREAL_1:185; ((p2 `2) / |.p2.|) - sn <= 0 by A82, XREAL_1:47; then - 1 <= (- (((p2 `2) / |.p2.|) - sn)) / (1 + sn) by A84; then ((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2 <= 1 ^2 by A90, SQUARE_1:49; then 1 - (((- (((p2 `2) / |.p2.|) - sn)) / (1 + sn)) ^2) >= 0 by XREAL_1:48; then A91: 1 - ((- ((((p2 `2) / |.p2.|) - sn) / (1 + sn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `1 = |.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))) by EUCLID:52; then A92: (|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `1) ^2 = (|.p2.| ^2) * ((sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2))) ^2) .= (|.p2.| ^2) * (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)) by A91, SQUARE_1:def_2 ; |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|.| ^2 = ((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `1) ^2) + ((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| `2) ^2) by JGRAPH_3:1 .= |.p2.| ^2 by A89, A92 ; then A93: sqrt (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|.| ^2) = |.p2.| by SQUARE_1:22; then A94: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]|.| = |.p2.| by SQUARE_1:22; |.|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]|.| ^2 = ((|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `1) ^2) + ((|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]| `2) ^2) by JGRAPH_3:1 .= |.p1.| ^2 by A88, A87 ; then A95: sqrt (|.|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]|.| ^2) = |.p1.| by SQUARE_1:22; then A96: |.|[(|.p1.| * (sqrt (1 - (((((p1 `2) / |.p1.|) - sn) / (1 + sn)) ^2)))),(|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn)))]|.| = |.p1.| by SQUARE_1:22; A97: (sn -FanMorphE) . p2 = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) - sn) / (1 + sn)) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) - sn) / (1 + sn)))]| by A1, A2, A82, Th84; then (((p2 `2) / |.p2.|) - sn) / (1 + sn) = (|.p1.| * ((((p1 `2) / |.p1.|) - sn) / (1 + sn))) / |.p2.| by A5, A68, A67, A82, A89, TOPRNS_1:24, XCMPLX_1:89; then (((p2 `2) / |.p2.|) - sn) / (1 + sn) = (((p1 `2) / |.p1.|) - sn) / (1 + sn) by A5, A68, A82, A97, A95, A93, TOPRNS_1:24, XCMPLX_1:89; then ((((p2 `2) / |.p2.|) - sn) / (1 + sn)) * (1 + sn) = ((p1 `2) / |.p1.|) - sn by A84, XCMPLX_1:87; then ((p2 `2) / |.p2.|) - sn = ((p1 `2) / |.p1.|) - sn by A84, XCMPLX_1:87; then ((p2 `2) / |.p2.|) * |.p2.| = p1 `2 by A5, A68, A82, A97, A96, A94, TOPRNS_1:24, XCMPLX_1:87; then A98: p2 `2 = p1 `2 by A82, TOPRNS_1:24, XCMPLX_1:87; A99: p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53; ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1; then p2 `1 = sqrt ((p1 `1) ^2) by A5, A68, A82, A97, A96, A94, A98, SQUARE_1:22; then p2 `1 = p1 `1 by A64, SQUARE_1:22; hence x1 = x2 by A98, A99, EUCLID:53; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; hence sn -FanMorphE is one-to-one by FUNCT_1:def_4; ::_thesis: verum end; theorem Th103: :: JGRAPH_4:103 for sn being Real st - 1 < sn & sn < 1 holds ( sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) ) proof let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies ( sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) ) ) assume that A1: - 1 < sn and A2: sn < 1 ; ::_thesis: ( sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) ) thus sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2) ; ::_thesis: rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = sn -FanMorphE holds rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) proof let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = sn -FanMorphE implies rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) ) assume A3: f = sn -FanMorphE ; ::_thesis: rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) A4: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; the carrier of (TOP-REAL 2) c= rng f proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (TOP-REAL 2) or y in rng f ) assume y in the carrier of (TOP-REAL 2) ; ::_thesis: y in rng f then reconsider p2 = y as Point of (TOP-REAL 2) ; set q = p2; now__::_thesis:_ex_x_being_set_st_ (_x_in_dom_(sn_-FanMorphE)_&_y_=_(sn_-FanMorphE)_._x_) percases ( p2 `1 <= 0 or ( (p2 `2) / |.p2.| >= 0 & p2 `1 >= 0 & p2 <> 0. (TOP-REAL 2) ) or ( (p2 `2) / |.p2.| < 0 & p2 `1 >= 0 & p2 <> 0. (TOP-REAL 2) ) ) by JGRAPH_2:3; suppose p2 `1 <= 0 ; ::_thesis: ex x being set st ( x in dom (sn -FanMorphE) & y = (sn -FanMorphE) . x ) then y = (sn -FanMorphE) . p2 by Th82; hence ex x being set st ( x in dom (sn -FanMorphE) & y = (sn -FanMorphE) . x ) by A3, A4; ::_thesis: verum end; supposeA5: ( (p2 `2) / |.p2.| >= 0 & p2 `1 >= 0 & p2 <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st ( x in dom (sn -FanMorphE) & y = (sn -FanMorphE) . x ) - (- (1 + sn)) > 0 by A1, XREAL_1:148; then A6: - ((- 1) - sn) > 0 ; A7: 1 - sn >= 0 by A2, XREAL_1:149; then ((p2 `2) / |.p2.|) * (1 - sn) >= 0 by A5; then (- 1) - sn <= ((p2 `2) / |.p2.|) * (1 - sn) by A6; then A8: ((- 1) - sn) + sn <= (((p2 `2) / |.p2.|) * (1 - sn)) + sn by XREAL_1:7; set px = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|; A9: |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn) by EUCLID:52; |.p2.| <> 0 by A5, TOPRNS_1:24; then A10: |.p2.| ^2 > 0 by SQUARE_1:12; A11: dom (sn -FanMorphE) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A12: 1 - sn > 0 by A2, XREAL_1:149; 0 <= (p2 `1) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A10, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (p2 `2) / |.p2.| <= 1 by SQUARE_1:51; then ((p2 `2) / |.p2.|) * (1 - sn) <= 1 * (1 - sn) by A12, XREAL_1:64; then ((((p2 `2) / |.p2.|) * (1 - sn)) + sn) - sn <= 1 - sn ; then (((p2 `2) / |.p2.|) * (1 - sn)) + sn <= 1 by XREAL_1:9; then 1 ^2 >= ((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2 by A8, SQUARE_1:49; then A13: 1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2) >= 0 by XREAL_1:48; then A14: sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)) >= 0 by SQUARE_1:def_2; A15: |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `1 = |.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))) by EUCLID:52; then |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 = ((|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))) ^2) + ((|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn)) ^2) by A9, JGRAPH_3:1 .= ((|.p2.| ^2) * ((sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))) ^2)) + ((|.p2.| ^2) * (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)) ; then A16: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2 = ((|.p2.| ^2) * (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2))) + ((|.p2.| ^2) * (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)) by A13, SQUARE_1:def_2 .= |.p2.| ^2 ; then A17: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| = sqrt (|.p2.| ^2) by SQUARE_1:22 .= |.p2.| by SQUARE_1:22 ; then A18: |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| <> 0. (TOP-REAL 2) by A5, TOPRNS_1:23, TOPRNS_1:24; (((p2 `2) / |.p2.|) * (1 - sn)) + sn >= 0 + sn by A5, A7, XREAL_1:7; then (|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| >= sn by A5, A9, A17, TOPRNS_1:24, XCMPLX_1:89; then A19: (sn -FanMorphE) . |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| = |[(|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (1 - (((((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) ^2)))),(|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * ((((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)))]| by A1, A2, A15, A14, A18, Th84; A20: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((p2 `1) / |.p2.|) ^2)) = |.p2.| * ((p2 `1) / |.p2.|) by A5, A17, SQUARE_1:22 .= p2 `1 by A5, TOPRNS_1:24, XCMPLX_1:87 ; A21: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * ((((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) = |.p2.| * ((((((p2 `2) / |.p2.|) * (1 - sn)) + sn) - sn) / (1 - sn)) by A5, A9, A17, TOPRNS_1:24, XCMPLX_1:89 .= |.p2.| * ((p2 `2) / |.p2.|) by A12, XCMPLX_1:89 .= p2 `2 by A5, TOPRNS_1:24, XCMPLX_1:87 ; then |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (1 - (((((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) - sn) / (1 - sn)) ^2))) = |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (1 - (((p2 `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.|) ^2))) by A5, A17, TOPRNS_1:24, XCMPLX_1:89 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (1 - (((p2 `2) ^2) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2)))) by XCMPLX_1:76 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2)) - (((p2 `2) ^2) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2)))) by A10, A16, XCMPLX_1:60 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2) - ((p2 `2) ^2)) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2))) by XCMPLX_1:120 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((((p2 `1) ^2) + ((p2 `2) ^2)) - ((p2 `2) ^2)) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| ^2))) by A16, JGRAPH_3:1 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 - sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 - sn)) + sn))]|.| * (sqrt (((p2 `1) / |.p2.|) ^2)) by A17, XCMPLX_1:76 ; hence ex x being set st ( x in dom (sn -FanMorphE) & y = (sn -FanMorphE) . x ) by A19, A21, A20, A11, EUCLID:53; ::_thesis: verum end; supposeA22: ( (p2 `2) / |.p2.| < 0 & p2 `1 >= 0 & p2 <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st ( x in dom (sn -FanMorphE) & y = (sn -FanMorphE) . x ) A23: 1 + sn >= 0 by A1, XREAL_1:148; 1 - sn > 0 by A2, XREAL_1:149; then A24: (1 - sn) + sn >= (((p2 `2) / |.p2.|) * (1 + sn)) + sn by A22, A23, XREAL_1:7; A25: 1 + sn > 0 by A1, XREAL_1:148; |.p2.| <> 0 by A22, TOPRNS_1:24; then A26: |.p2.| ^2 > 0 by SQUARE_1:12; 0 <= (p2 `1) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `2) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p2 `2) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; then ((p2 `2) ^2) / (|.p2.| ^2) <= 1 by A26, XCMPLX_1:60; then ((p2 `2) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (p2 `2) / |.p2.| >= - 1 by SQUARE_1:51; then ((p2 `2) / |.p2.|) * (1 + sn) >= (- 1) * (1 + sn) by A25, XREAL_1:64; then ((((p2 `2) / |.p2.|) * (1 + sn)) + sn) - sn >= (- 1) - sn ; then (((p2 `2) / |.p2.|) * (1 + sn)) + sn >= - 1 by XREAL_1:9; then 1 ^2 >= ((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2 by A24, SQUARE_1:49; then A27: 1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2) >= 0 by XREAL_1:48; then A28: sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)) >= 0 by SQUARE_1:def_2; A29: dom (sn -FanMorphE) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; set px = |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|; A30: |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2 = |.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn) by EUCLID:52; A31: |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `1 = |.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))) by EUCLID:52; then |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 = ((|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))) ^2) + ((|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn)) ^2) by A30, JGRAPH_3:1 .= ((|.p2.| ^2) * ((sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))) ^2)) + ((|.p2.| ^2) * (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)) ; then A32: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2 = ((|.p2.| ^2) * (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2))) + ((|.p2.| ^2) * (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)) by A27, SQUARE_1:def_2 .= |.p2.| ^2 ; then A33: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| = sqrt (|.p2.| ^2) by SQUARE_1:22 .= |.p2.| by SQUARE_1:22 ; then A34: |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| <> 0. (TOP-REAL 2) by A22, TOPRNS_1:23, TOPRNS_1:24; (((p2 `2) / |.p2.|) * (1 + sn)) + sn <= 0 + sn by A22, A23, XREAL_1:7; then (|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| <= sn by A22, A30, A33, TOPRNS_1:24, XCMPLX_1:89; then A35: (sn -FanMorphE) . |[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| = |[(|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (1 - (((((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) ^2)))),(|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * ((((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)))]| by A1, A2, A31, A28, A34, Th84; A36: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((p2 `1) / |.p2.|) ^2)) = |.p2.| * ((p2 `1) / |.p2.|) by A22, A33, SQUARE_1:22 .= p2 `1 by A22, TOPRNS_1:24, XCMPLX_1:87 ; A37: |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * ((((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) = |.p2.| * ((((((p2 `2) / |.p2.|) * (1 + sn)) + sn) - sn) / (1 + sn)) by A22, A30, A33, TOPRNS_1:24, XCMPLX_1:89 .= |.p2.| * ((p2 `2) / |.p2.|) by A25, XCMPLX_1:89 .= p2 `2 by A22, TOPRNS_1:24, XCMPLX_1:87 ; then |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (1 - (((((|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]| `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) - sn) / (1 + sn)) ^2))) = |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (1 - (((p2 `2) / |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.|) ^2))) by A22, A33, TOPRNS_1:24, XCMPLX_1:89 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (1 - (((p2 `2) ^2) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2)))) by XCMPLX_1:76 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2)) - (((p2 `2) ^2) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2)))) by A26, A32, XCMPLX_1:60 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2) - ((p2 `2) ^2)) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2))) by XCMPLX_1:120 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((((p2 `1) ^2) + ((p2 `2) ^2)) - ((p2 `2) ^2)) / (|.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| ^2))) by A32, JGRAPH_3:1 .= |.|[(|.p2.| * (sqrt (1 - (((((p2 `2) / |.p2.|) * (1 + sn)) + sn) ^2)))),(|.p2.| * ((((p2 `2) / |.p2.|) * (1 + sn)) + sn))]|.| * (sqrt (((p2 `1) / |.p2.|) ^2)) by A33, XCMPLX_1:76 ; hence ex x being set st ( x in dom (sn -FanMorphE) & y = (sn -FanMorphE) . x ) by A35, A37, A36, A29, EUCLID:53; ::_thesis: verum end; end; end; hence y in rng f by A3, FUNCT_1:def_3; ::_thesis: verum end; hence rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) by A3, XBOOLE_0:def_10; ::_thesis: verum end; hence rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) ; ::_thesis: verum end; theorem Th104: :: JGRAPH_4:104 for sn being Real for p2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 holds ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphE) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE) . p2 in V2 ) ) proof reconsider O = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:67; let sn be Real; ::_thesis: for p2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 holds ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphE) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE) . p2 in V2 ) ) let p2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 implies ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphE) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE) . p2 in V2 ) ) ) A1: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider V0 = Ball (O,(|.p2.| + 1)) as Subset of (TOP-REAL 2) ; ( O in V0 & V0 c= cl_Ball (O,(|.p2.| + 1)) ) by GOBOARD6:1, METRIC_1:14; then reconsider K0 = cl_Ball (O,(|.p2.| + 1)) as non empty compact Subset of (TOP-REAL 2) by A1, Th15; set q3 = (sn -FanMorphE) . p2; reconsider VV0 = V0 as Subset of (TopSpaceMetr (Euclid 2)) ; reconsider u2 = p2 as Point of (Euclid 2) by EUCLID:67; reconsider u3 = (sn -FanMorphE) . p2 as Point of (Euclid 2) by EUCLID:67; A2: (sn -FanMorphE) .: K0 c= K0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (sn -FanMorphE) .: K0 or y in K0 ) assume y in (sn -FanMorphE) .: K0 ; ::_thesis: y in K0 then consider x being set such that A3: x in dom (sn -FanMorphE) and A4: x in K0 and A5: y = (sn -FanMorphE) . x by FUNCT_1:def_6; reconsider q = x as Point of (TOP-REAL 2) by A3; reconsider uq = q as Point of (Euclid 2) by EUCLID:67; dist (O,uq) <= |.p2.| + 1 by A4, METRIC_1:12; then |.((0. (TOP-REAL 2)) - q).| <= |.p2.| + 1 by JGRAPH_1:28; then |.(- q).| <= |.p2.| + 1 by EUCLID:27; then A6: |.q.| <= |.p2.| + 1 by TOPRNS_1:26; A7: y in rng (sn -FanMorphE) by A3, A5, FUNCT_1:def_3; then reconsider u = y as Point of (Euclid 2) by EUCLID:67; reconsider q4 = y as Point of (TOP-REAL 2) by A7; |.q4.| = |.q.| by A5, Th97; then |.(- q4).| <= |.p2.| + 1 by A6, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - q4).| <= |.p2.| + 1 by EUCLID:27; then dist (O,u) <= |.p2.| + 1 by JGRAPH_1:28; hence y in K0 by METRIC_1:12; ::_thesis: verum end; VV0 is open by TOPMETR:14; then A8: V0 is open by Lm11, PRE_TOPC:30; A9: |.p2.| < |.p2.| + 1 by XREAL_1:29; then |.(- p2).| < |.p2.| + 1 by TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - p2).| < |.p2.| + 1 by EUCLID:27; then dist (O,u2) < |.p2.| + 1 by JGRAPH_1:28; then A10: p2 in V0 by METRIC_1:11; |.((sn -FanMorphE) . p2).| = |.p2.| by Th97; then |.(- ((sn -FanMorphE) . p2)).| < |.p2.| + 1 by A9, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - ((sn -FanMorphE) . p2)).| < |.p2.| + 1 by EUCLID:27; then dist (O,u3) < |.p2.| + 1 by JGRAPH_1:28; then A11: (sn -FanMorphE) . p2 in V0 by METRIC_1:11; assume A12: ( - 1 < sn & sn < 1 ) ; ::_thesis: ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphE) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE) . p2 in V2 ) ) K0 c= (sn -FanMorphE) .: K0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in K0 or y in (sn -FanMorphE) .: K0 ) assume A13: y in K0 ; ::_thesis: y in (sn -FanMorphE) .: K0 then reconsider q4 = y as Point of (TOP-REAL 2) ; reconsider y = y as Point of (Euclid 2) by A13; the carrier of (TOP-REAL 2) c= rng (sn -FanMorphE) by A12, Th103; then q4 in rng (sn -FanMorphE) by TARSKI:def_3; then consider x being set such that A14: x in dom (sn -FanMorphE) and A15: y = (sn -FanMorphE) . x by FUNCT_1:def_3; reconsider x = x as Point of (Euclid 2) by A14, Lm11; reconsider q = x as Point of (TOP-REAL 2) by A14; |.q4.| = |.q.| by A15, Th97; then q in K0 by A13, Lm12; hence y in (sn -FanMorphE) .: K0 by A14, A15, FUNCT_1:def_6; ::_thesis: verum end; then K0 = (sn -FanMorphE) .: K0 by A2, XBOOLE_0:def_10; hence ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (sn -FanMorphE) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE) . p2 in V2 ) ) by A10, A8, A11, METRIC_1:14; ::_thesis: verum end; theorem :: JGRAPH_4:105 for sn being Real st - 1 < sn & sn < 1 holds ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = sn -FanMorphE & f is being_homeomorphism ) proof let sn be Real; ::_thesis: ( - 1 < sn & sn < 1 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = sn -FanMorphE & f is being_homeomorphism ) ) reconsider f = sn -FanMorphE as Function of (TOP-REAL 2),(TOP-REAL 2) ; assume A1: ( - 1 < sn & sn < 1 ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = sn -FanMorphE & f is being_homeomorphism ) then A2: for p2 being Point of (TOP-REAL 2) ex K being non empty compact Subset of (TOP-REAL 2) st ( K = f .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & f . p2 in V2 ) ) by Th104; ( rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) & ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = sn -FanMorphE & h is continuous ) ) by A1, Th101, Th103; then f is being_homeomorphism by A1, A2, Th3, Th102; hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = sn -FanMorphE & f is being_homeomorphism ) ; ::_thesis: verum end; theorem Th106: :: JGRAPH_4:106 for sn being Real for q being Point of (TOP-REAL 2) st sn < 1 & q `1 > 0 & (q `2) / |.q.| >= sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 >= 0 ) proof let sn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st sn < 1 & q `1 > 0 & (q `2) / |.q.| >= sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 >= 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( sn < 1 & q `1 > 0 & (q `2) / |.q.| >= sn implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 >= 0 ) ) assume that A1: sn < 1 and A2: q `1 > 0 and A3: (q `2) / |.q.| >= sn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 >= 0 ) A4: ((q `2) / |.q.|) - sn >= 0 by A3, XREAL_1:48; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (sn -FanMorphE) . q implies ( p `1 > 0 & p `2 >= 0 ) ) set qz = p; A5: 1 - sn > 0 by A1, XREAL_1:149; A6: |.q.| <> 0 by A2, JGRAPH_2:3, TOPRNS_1:24; then A7: |.q.| ^2 > 0 by SQUARE_1:12; ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) < ((q `1) ^2) + ((q `2) ^2) ) by A2, JGRAPH_3:1, SQUARE_1:12, XREAL_1:8; then ((q `2) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A7, XREAL_1:74; then ((q `2) ^2) / (|.q.| ^2) < 1 by A7, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 < 1 by XCMPLX_1:76; then 1 > (q `2) / |.q.| by SQUARE_1:52; then 1 - sn > ((q `2) / |.q.|) - sn by XREAL_1:9; then - (1 - sn) < - (((q `2) / |.q.|) - sn) by XREAL_1:24; then (- (1 - sn)) / (1 - sn) < (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A5, XREAL_1:74; then - 1 < (- (((q `2) / |.q.|) - sn)) / (1 - sn) by A5, XCMPLX_1:197; then ((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2 < 1 ^2 by A5, A4, SQUARE_1:50; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2)) > 0 by SQUARE_1:25; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) ^2) / ((1 - sn) ^2))) > 0 by XCMPLX_1:76; then sqrt (1 - (((((q `2) / |.q.|) - sn) ^2) / ((1 - sn) ^2))) > 0 ; then A8: sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)) > 0 by XCMPLX_1:76; assume p = (sn -FanMorphE) . q ; ::_thesis: ( p `1 > 0 & p `2 >= 0 ) then A9: p = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by A2, A3, Th82; then p `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) by EUCLID:52; hence ( p `1 > 0 & p `2 >= 0 ) by A9, A6, A5, A4, A8, EUCLID:52, XREAL_1:129; ::_thesis: verum end; theorem Th107: :: JGRAPH_4:107 for sn being Real for q being Point of (TOP-REAL 2) st - 1 < sn & q `1 > 0 & (q `2) / |.q.| < sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 < 0 ) proof let sn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < sn & q `1 > 0 & (q `2) / |.q.| < sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 < 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & q `1 > 0 & (q `2) / |.q.| < sn implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 < 0 ) ) assume that A1: - 1 < sn and A2: q `1 > 0 and A3: (q `2) / |.q.| < sn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 < 0 ) A4: 1 + sn > 0 by A1, XREAL_1:148; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (sn -FanMorphE) . q implies ( p `1 > 0 & p `2 < 0 ) ) set qz = p; assume p = (sn -FanMorphE) . q ; ::_thesis: ( p `1 > 0 & p `2 < 0 ) then p = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| by A2, A3, Th83; then A5: ( p `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))) & p `2 = |.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)) ) by EUCLID:52; A6: |.q.| <> 0 by A2, JGRAPH_2:3, TOPRNS_1:24; then A7: |.q.| ^2 > 0 by SQUARE_1:12; A8: ((q `2) / |.q.|) - sn < 0 by A3, XREAL_1:49; then - (((q `2) / |.q.|) - sn) > 0 by XREAL_1:58; then (- (1 + sn)) / (1 + sn) < (- (((q `2) / |.q.|) - sn)) / (1 + sn) by A4, XREAL_1:74; then A9: - 1 < (- (((q `2) / |.q.|) - sn)) / (1 + sn) by A4, XCMPLX_1:197; ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `2) ^2) < ((q `1) ^2) + ((q `2) ^2) ) by A2, JGRAPH_3:1, SQUARE_1:12, XREAL_1:8; then ((q `2) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A7, XREAL_1:74; then ((q `2) ^2) / (|.q.| ^2) < 1 by A7, XCMPLX_1:60; then ((q `2) / |.q.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (q `2) / |.q.| by SQUARE_1:52; then (- 1) - sn < ((q `2) / |.q.|) - sn by XREAL_1:9; then - (- (1 + sn)) > - (((q `2) / |.q.|) - sn) by XREAL_1:24; then (- (((q `2) / |.q.|) - sn)) / (1 + sn) < 1 by A4, XREAL_1:191; then ((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2 < 1 ^2 by A9, SQUARE_1:50; then 1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) / (1 + sn)) ^2)) > 0 by SQUARE_1:25; then sqrt (1 - (((- (((q `2) / |.q.|) - sn)) ^2) / ((1 + sn) ^2))) > 0 by XCMPLX_1:76; then sqrt (1 - (((((q `2) / |.q.|) - sn) ^2) / ((1 + sn) ^2))) > 0 ; then A10: sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)) > 0 by XCMPLX_1:76; (((q `2) / |.q.|) - sn) / (1 + sn) < 0 by A1, A8, XREAL_1:141, XREAL_1:148; hence ( p `1 > 0 & p `2 < 0 ) by A6, A5, A10, XREAL_1:129, XREAL_1:132; ::_thesis: verum end; theorem Th108: :: JGRAPH_4:108 for sn being Real for q1, q2 being Point of (TOP-REAL 2) st sn < 1 & q1 `1 > 0 & (q1 `2) / |.q1.| >= sn & q2 `1 > 0 & (q2 `2) / |.q2.| >= sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| proof let sn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st sn < 1 & q1 `1 > 0 & (q1 `2) / |.q1.| >= sn & q2 `1 > 0 & (q2 `2) / |.q2.| >= sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( sn < 1 & q1 `1 > 0 & (q1 `2) / |.q1.| >= sn & q2 `1 > 0 & (q2 `2) / |.q2.| >= sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A1: sn < 1 and A2: q1 `1 > 0 and A3: (q1 `2) / |.q1.| >= sn and A4: q2 `1 > 0 and A5: (q2 `2) / |.q2.| >= sn and A6: (q1 `2) / |.q1.| < (q2 `2) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| A7: ( ((q1 `2) / |.q1.|) - sn < ((q2 `2) / |.q2.|) - sn & 1 - sn > 0 ) by A1, A6, XREAL_1:9, XREAL_1:149; let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 implies (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A8: p1 = (sn -FanMorphE) . q1 and A9: p2 = (sn -FanMorphE) . q2 ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| A10: |.p2.| = |.q2.| by A9, Th97; p2 = |[(|.q2.| * (sqrt (1 - (((((q2 `2) / |.q2.|) - sn) / (1 - sn)) ^2)))),(|.q2.| * ((((q2 `2) / |.q2.|) - sn) / (1 - sn)))]| by A4, A5, A9, Th82; then A11: p2 `2 = |.q2.| * ((((q2 `2) / |.q2.|) - sn) / (1 - sn)) by EUCLID:52; |.q2.| > 0 by A4, Lm1, JGRAPH_2:3; then A12: (p2 `2) / |.p2.| = (((q2 `2) / |.q2.|) - sn) / (1 - sn) by A11, A10, XCMPLX_1:89; p1 = |[(|.q1.| * (sqrt (1 - (((((q1 `2) / |.q1.|) - sn) / (1 - sn)) ^2)))),(|.q1.| * ((((q1 `2) / |.q1.|) - sn) / (1 - sn)))]| by A2, A3, A8, Th82; then A13: p1 `2 = |.q1.| * ((((q1 `2) / |.q1.|) - sn) / (1 - sn)) by EUCLID:52; A14: |.p1.| = |.q1.| by A8, Th97; |.q1.| > 0 by A2, Lm1, JGRAPH_2:3; then (p1 `2) / |.p1.| = (((q1 `2) / |.q1.|) - sn) / (1 - sn) by A13, A14, XCMPLX_1:89; hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A12, A7, XREAL_1:74; ::_thesis: verum end; theorem Th109: :: JGRAPH_4:109 for sn being Real for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & q1 `1 > 0 & (q1 `2) / |.q1.| < sn & q2 `1 > 0 & (q2 `2) / |.q2.| < sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| proof let sn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & q1 `1 > 0 & (q1 `2) / |.q1.| < sn & q2 `1 > 0 & (q2 `2) / |.q2.| < sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & q1 `1 > 0 & (q1 `2) / |.q1.| < sn & q2 `1 > 0 & (q2 `2) / |.q2.| < sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A1: - 1 < sn and A2: q1 `1 > 0 and A3: (q1 `2) / |.q1.| < sn and A4: q2 `1 > 0 and A5: (q2 `2) / |.q2.| < sn and A6: (q1 `2) / |.q1.| < (q2 `2) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| A7: ( ((q1 `2) / |.q1.|) - sn < ((q2 `2) / |.q2.|) - sn & 1 + sn > 0 ) by A1, A6, XREAL_1:9, XREAL_1:148; let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 implies (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A8: p1 = (sn -FanMorphE) . q1 and A9: p2 = (sn -FanMorphE) . q2 ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| A10: |.p2.| = |.q2.| by A9, Th97; p2 = |[(|.q2.| * (sqrt (1 - (((((q2 `2) / |.q2.|) - sn) / (1 + sn)) ^2)))),(|.q2.| * ((((q2 `2) / |.q2.|) - sn) / (1 + sn)))]| by A4, A5, A9, Th83; then A11: p2 `2 = |.q2.| * ((((q2 `2) / |.q2.|) - sn) / (1 + sn)) by EUCLID:52; |.q2.| > 0 by A4, Lm1, JGRAPH_2:3; then A12: (p2 `2) / |.p2.| = (((q2 `2) / |.q2.|) - sn) / (1 + sn) by A11, A10, XCMPLX_1:89; p1 = |[(|.q1.| * (sqrt (1 - (((((q1 `2) / |.q1.|) - sn) / (1 + sn)) ^2)))),(|.q1.| * ((((q1 `2) / |.q1.|) - sn) / (1 + sn)))]| by A2, A3, A8, Th83; then A13: p1 `2 = |.q1.| * ((((q1 `2) / |.q1.|) - sn) / (1 + sn)) by EUCLID:52; A14: |.p1.| = |.q1.| by A8, Th97; |.q1.| > 0 by A2, Lm1, JGRAPH_2:3; then (p1 `2) / |.p1.| = (((q1 `2) / |.q1.|) - sn) / (1 + sn) by A13, A14, XCMPLX_1:89; hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A12, A7, XREAL_1:74; ::_thesis: verum end; theorem :: JGRAPH_4:110 for sn being Real for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q1 `1 > 0 & q2 `1 > 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| proof let sn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q1 `1 > 0 & q2 `1 > 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 & q1 `1 > 0 & q2 `1 > 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A1: - 1 < sn and A2: sn < 1 and A3: q1 `1 > 0 and A4: q2 `1 > 0 and A5: (q1 `2) / |.q1.| < (q2 `2) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds (p1 `2) / |.p1.| < (p2 `2) / |.p2.| let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 implies (p1 `2) / |.p1.| < (p2 `2) / |.p2.| ) assume that A6: p1 = (sn -FanMorphE) . q1 and A7: p2 = (sn -FanMorphE) . q2 ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| percases ( ( (q1 `2) / |.q1.| >= sn & (q2 `2) / |.q2.| >= sn ) or ( (q1 `2) / |.q1.| >= sn & (q2 `2) / |.q2.| < sn ) or ( (q1 `2) / |.q1.| < sn & (q2 `2) / |.q2.| >= sn ) or ( (q1 `2) / |.q1.| < sn & (q2 `2) / |.q2.| < sn ) ) ; suppose ( (q1 `2) / |.q1.| >= sn & (q2 `2) / |.q2.| >= sn ) ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A2, A3, A4, A5, A6, A7, Th108; ::_thesis: verum end; suppose ( (q1 `2) / |.q1.| >= sn & (q2 `2) / |.q2.| < sn ) ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A5, XXREAL_0:2; ::_thesis: verum end; supposeA8: ( (q1 `2) / |.q1.| < sn & (q2 `2) / |.q2.| >= sn ) ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| then p2 `2 >= 0 by A2, A4, A7, Th106; then A9: (p2 `2) / |.p2.| >= 0 ; p1 `2 < 0 by A1, A3, A6, A8, Th107; hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A9, Lm1, JGRAPH_2:3, XREAL_1:141; ::_thesis: verum end; suppose ( (q1 `2) / |.q1.| < sn & (q2 `2) / |.q2.| < sn ) ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.| hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A1, A3, A4, A5, A6, A7, Th109; ::_thesis: verum end; end; end; theorem :: JGRAPH_4:111 for sn being Real for q being Point of (TOP-REAL 2) st q `1 > 0 & (q `2) / |.q.| = sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 = 0 ) proof let sn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st q `1 > 0 & (q `2) / |.q.| = sn holds for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 = 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( q `1 > 0 & (q `2) / |.q.| = sn implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 = 0 ) ) assume that A1: q `1 > 0 and A2: (q `2) / |.q.| = sn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds ( p `1 > 0 & p `2 = 0 ) A3: ( |.q.| <> 0 & sqrt (1 - (((- (((q `2) / |.q.|) - sn)) / (1 - sn)) ^2)) > 0 ) by A1, A2, JGRAPH_2:3, SQUARE_1:25, TOPRNS_1:24; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (sn -FanMorphE) . q implies ( p `1 > 0 & p `2 = 0 ) ) assume p = (sn -FanMorphE) . q ; ::_thesis: ( p `1 > 0 & p `2 = 0 ) then A4: p = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| by A1, A2, Th82; then p `1 = |.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))) by EUCLID:52; hence ( p `1 > 0 & p `2 = 0 ) by A2, A4, A3, EUCLID:52, XREAL_1:129; ::_thesis: verum end; theorem :: JGRAPH_4:112 for sn being real number holds 0. (TOP-REAL 2) = (sn -FanMorphE) . (0. (TOP-REAL 2)) by Th82, JGRAPH_2:3; begin definition let s be real number ; let q be Point of (TOP-REAL 2); func FanS (s,q) -> Point of (TOP-REAL 2) equals :Def8: :: JGRAPH_4:def 8 |.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2))))]| if ( (q `1) / |.q.| >= s & q `2 < 0 ) |.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2))))]| if ( (q `1) / |.q.| < s & q `2 < 0 ) otherwise q; correctness coherence ( ( (q `1) / |.q.| >= s & q `2 < 0 implies |.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2))))]| is Point of (TOP-REAL 2) ) & ( (q `1) / |.q.| < s & q `2 < 0 implies |.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2))))]| is Point of (TOP-REAL 2) ) & ( ( not (q `1) / |.q.| >= s or not q `2 < 0 ) & ( not (q `1) / |.q.| < s or not q `2 < 0 ) implies q is Point of (TOP-REAL 2) ) ); consistency for b1 being Point of (TOP-REAL 2) st (q `1) / |.q.| >= s & q `2 < 0 & (q `1) / |.q.| < s & q `2 < 0 holds ( b1 = |.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2))))]| iff b1 = |.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2))))]| ); ; end; :: deftheorem Def8 defines FanS JGRAPH_4:def_8_:_ for s being real number for q being Point of (TOP-REAL 2) holds ( ( (q `1) / |.q.| >= s & q `2 < 0 implies FanS (s,q) = |.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2))))]| ) & ( (q `1) / |.q.| < s & q `2 < 0 implies FanS (s,q) = |.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2))))]| ) & ( ( not (q `1) / |.q.| >= s or not q `2 < 0 ) & ( not (q `1) / |.q.| < s or not q `2 < 0 ) implies FanS (s,q) = q ) ); definition let c be real number ; funcc -FanMorphS -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def9: :: JGRAPH_4:def 9 for q being Point of (TOP-REAL 2) holds it . q = FanS (c,q); existence ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st for q being Point of (TOP-REAL 2) holds b1 . q = FanS (c,q) proof deffunc H1( Point of (TOP-REAL 2)) -> Point of (TOP-REAL 2) = FanS (c,$1); thus ex IT being Function of (TOP-REAL 2),(TOP-REAL 2) st for q being Point of (TOP-REAL 2) holds IT . q = H1(q) from FUNCT_2:sch_4(); ::_thesis: verum end; uniqueness for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for q being Point of (TOP-REAL 2) holds b1 . q = FanS (c,q) ) & ( for q being Point of (TOP-REAL 2) holds b2 . q = FanS (c,q) ) holds b1 = b2 proof deffunc H1( Point of (TOP-REAL 2)) -> Point of (TOP-REAL 2) = FanS (c,$1); thus for a, b being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for q being Point of (TOP-REAL 2) holds a . q = H1(q) ) & ( for q being Point of (TOP-REAL 2) holds b . q = H1(q) ) holds a = b from BINOP_2:sch_1(); ::_thesis: verum end; end; :: deftheorem Def9 defines -FanMorphS JGRAPH_4:def_9_:_ for c being real number for b2 being Function of (TOP-REAL 2),(TOP-REAL 2) holds ( b2 = c -FanMorphS iff for q being Point of (TOP-REAL 2) holds b2 . q = FanS (c,q) ); theorem Th113: :: JGRAPH_4:113 for q being Point of (TOP-REAL 2) for cn being real number holds ( ( (q `1) / |.q.| >= cn & q `2 < 0 implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( q `2 >= 0 implies (cn -FanMorphS) . q = q ) ) proof let q be Point of (TOP-REAL 2); ::_thesis: for cn being real number holds ( ( (q `1) / |.q.| >= cn & q `2 < 0 implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( q `2 >= 0 implies (cn -FanMorphS) . q = q ) ) let cn be real number ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 < 0 implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( q `2 >= 0 implies (cn -FanMorphS) . q = q ) ) hereby ::_thesis: ( q `2 >= 0 implies (cn -FanMorphS) . q = q ) assume ( (q `1) / |.q.| >= cn & q `2 < 0 ) ; ::_thesis: (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| then FanS (cn,q) = |.q.| * |[((((q `1) / |.q.|) - cn) / (1 - cn)),(- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by Def8 .= |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by EUCLID:58 ; hence (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by Def9; ::_thesis: verum end; assume A1: q `2 >= 0 ; ::_thesis: (cn -FanMorphS) . q = q (cn -FanMorphS) . q = FanS (cn,q) by Def9; hence (cn -FanMorphS) . q = q by A1, Def8; ::_thesis: verum end; theorem Th114: :: JGRAPH_4:114 for q being Point of (TOP-REAL 2) for cn being Real st (q `1) / |.q.| <= cn & q `2 < 0 holds (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| proof let q be Point of (TOP-REAL 2); ::_thesis: for cn being Real st (q `1) / |.q.| <= cn & q `2 < 0 holds (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| let cn be Real; ::_thesis: ( (q `1) / |.q.| <= cn & q `2 < 0 implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) assume that A1: (q `1) / |.q.| <= cn and A2: q `2 < 0 ; ::_thesis: (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| percases ( (q `1) / |.q.| < cn or (q `1) / |.q.| = cn ) by A1, XXREAL_0:1; suppose (q `1) / |.q.| < cn ; ::_thesis: (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| then FanS (cn,q) = |.q.| * |[((((q `1) / |.q.|) - cn) / (1 + cn)),(- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| by A2, Def8 .= |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| by EUCLID:58 ; hence (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| by Def9; ::_thesis: verum end; supposeA3: (q `1) / |.q.| = cn ; ::_thesis: (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| then (((q `1) / |.q.|) - cn) / (1 - cn) = 0 ; hence (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| by A2, A3, Th113; ::_thesis: verum end; end; end; theorem Th115: :: JGRAPH_4:115 for q being Point of (TOP-REAL 2) for cn being Real st - 1 < cn & cn < 1 holds ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) proof let q be Point of (TOP-REAL 2); ::_thesis: for cn being Real st - 1 < cn & cn < 1 holds ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) ) assume that A1: - 1 < cn and A2: cn < 1 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) percases ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) ) or ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) ) or q `2 > 0 or q = 0. (TOP-REAL 2) ) ; supposeA3: ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) percases ( q `2 < 0 or q `2 >= 0 ) ; supposeA4: q `2 < 0 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) then FanS (cn,q) = |.q.| * |[((((q `1) / |.q.|) - cn) / (1 - cn)),(- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| by A3, Def8 .= |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by EUCLID:58 ; hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) by A4, Def9, Th114; ::_thesis: verum end; supposeA5: q `2 >= 0 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) then A6: (cn -FanMorphS) . q = q by Th113; A7: |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) by JGRAPH_3:1; A8: 1 - cn > 0 by A2, XREAL_1:149; A9: q `2 = 0 by A3, A5; |.q.| <> 0 by A3, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `1) ^2) / (|.q.| ^2) = 1 ^2 by A7, A9, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 = 1 ^2 by XCMPLX_1:76; then A10: sqrt (((q `1) / |.q.|) ^2) = 1 by SQUARE_1:22; A11: now__::_thesis:_not_q_`1_<_0 assume q `1 < 0 ; ::_thesis: contradiction then - ((q `1) / |.q.|) = 1 by A10, SQUARE_1:23; hence contradiction by A1, A3; ::_thesis: verum end; sqrt (|.q.| ^2) = |.q.| by SQUARE_1:22; then A12: |.q.| = q `1 by A7, A9, A11, SQUARE_1:22; then 1 = (q `1) / |.q.| by A3, TOPRNS_1:24, XCMPLX_1:60; then (((q `1) / |.q.|) - cn) / (1 - cn) = 1 by A8, XCMPLX_1:60; hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) by A2, A6, A9, A12, EUCLID:53, SQUARE_1:17, TOPRNS_1:24, XCMPLX_1:60; ::_thesis: verum end; end; end; supposeA13: ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) percases ( q `2 < 0 or q `2 >= 0 ) ; suppose q `2 < 0 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) by Th113, Th114; ::_thesis: verum end; supposeA14: q `2 >= 0 ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) then A15: q `2 = 0 by A13; A16: 1 + cn > 0 by A1, XREAL_1:148; A17: |.q.| <> 0 by A13, TOPRNS_1:24; 1 > (q `1) / |.q.| by A2, A13, XXREAL_0:2; then 1 * |.q.| > ((q `1) / |.q.|) * |.q.| by A17, XREAL_1:68; then A18: ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & |.q.| > q `1 ) by A13, JGRAPH_3:1, TOPRNS_1:24, XCMPLX_1:87; then A19: |.q.| = - (q `1) by A15, SQUARE_1:40; A20: q `1 = - |.q.| by A15, A18, SQUARE_1:40; then - 1 = (q `1) / |.q.| by A13, TOPRNS_1:24, XCMPLX_1:197; then (((q `1) / |.q.|) - cn) / (1 + cn) = (- (1 + cn)) / (1 + cn) .= - 1 by A16, XCMPLX_1:197 ; then |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| = q by A15, A19, EUCLID:53, SQUARE_1:17; hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) by A1, A14, A17, A20, Th113, XCMPLX_1:197; ::_thesis: verum end; end; end; suppose ( q `2 > 0 or q = 0. (TOP-REAL 2) ) ; ::_thesis: ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) hence ( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) ) ; ::_thesis: verum end; end; end; theorem Th116: :: JGRAPH_4:116 for cn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let cn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = cn; set b = 1 - cn; reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2; assume that A1: cn < 1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 - cn > 0 by A1, XREAL_1:149; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (((r1 / r2) - cn) / (1 - cn)) and A6: g3 is continuous by A4, Th5; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in K1 by A7, A8, A9, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( proj1 . r = r `1 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_5; A11: ( g2 . s = proj1 . s & g1 . s = (2 NormF) . s ) by Lm2, Lm5; f . r = |.r.| * ((((r `1) / |.r.|) - cn) / (1 - cn)) by A2, A9; hence f . x = g3 . x by A5, A11, A10; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th117: :: JGRAPH_4:117 for cn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let cn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = cn; set b = 1 + cn; reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2; assume that A1: - 1 < cn and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 + cn > 0 by A1, XREAL_1:148; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being Real st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (((r1 / r2) - cn) / (1 + cn)) and A6: g3 is continuous by A4, Th5; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; A8: for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then reconsider s = x as Point of ((TOP-REAL 2) | K1) ; x in dom g3 by A7, A9; then x in K1 by A7, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A10: ( proj1 . r = r `1 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_5; A11: ( g2 . s = proj1 . s & g1 . s = (2 NormF) . s ) by Lm2, Lm5; f . r = |.r.| * ((((r `1) / |.r.|) - cn) / (1 + cn)) by A2, A9; hence f . x = g3 . x by A5, A11, A10; ::_thesis: verum end; dom f = dom g3 by A7, FUNCT_2:def_1; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th118: :: JGRAPH_4:118 for cn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let cn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = cn; set b = 1 - cn; reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2; assume that A1: cn < 1 and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; 1 - cn > 0 by A1, XREAL_1:149; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - cn) / (1 - cn)) ^2))))) and A6: g3 is continuous by A4, Th9; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) A9: 1 - cn > 0 by A1, XREAL_1:149; assume A10: x in dom f ; ::_thesis: f . x = g3 . x then x in K1 by A7, A8, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; A11: |.r.| <> 0 by A3, A10, TOPRNS_1:24; |.r.| ^2 = ((r `1) ^2) + ((r `2) ^2) by JGRAPH_3:1; then A12: ((r `1) - |.r.|) * ((r `1) + |.r.|) = - ((r `2) ^2) ; (r `2) ^2 >= 0 by XREAL_1:63; then r `1 <= |.r.| by A12, XREAL_1:93; then (r `1) / |.r.| <= |.r.| / |.r.| by XREAL_1:72; then (r `1) / |.r.| <= 1 by A11, XCMPLX_1:60; then A13: ((r `1) / |.r.|) - cn <= 1 - cn by XREAL_1:9; reconsider s = x as Point of ((TOP-REAL 2) | K1) by A10; A14: now__::_thesis:_not_(1_-_cn)_^2_=_0 assume (1 - cn) ^2 = 0 ; ::_thesis: contradiction then (1 - cn) + cn = 0 + cn by XCMPLX_1:6; hence contradiction by A1; ::_thesis: verum end; cn - ((r `1) / |.r.|) <= 0 by A3, A10, XREAL_1:47; then - (cn - ((r `1) / |.r.|)) >= - (1 - cn) by A9, XREAL_1:24; then ( (1 - cn) ^2 >= 0 & (((r `1) / |.r.|) - cn) ^2 <= (1 - cn) ^2 ) by A13, SQUARE_1:49, XREAL_1:63; then ((((r `1) / |.r.|) - cn) ^2) / ((1 - cn) ^2) <= ((1 - cn) ^2) / ((1 - cn) ^2) by XREAL_1:72; then ((((r `1) / |.r.|) - cn) ^2) / ((1 - cn) ^2) <= 1 by A14, XCMPLX_1:60; then ((((r `1) / |.r.|) - cn) / (1 - cn)) ^2 <= 1 by XCMPLX_1:76; then 1 - (((((r `1) / |.r.|) - cn) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then abs (1 - (((((r `1) / |.r.|) - cn) / (1 - cn)) ^2)) = 1 - (((((r `1) / |.r.|) - cn) / (1 - cn)) ^2) by ABSVALUE:def_1; then A15: f . r = |.r.| * (- (sqrt (abs (1 - (((((r `1) / |.r.|) - cn) / (1 - cn)) ^2))))) by A2, A10; A16: ( proj1 . r = r `1 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_5; ( g2 . s = proj1 . s & g1 . s = (2 NormF) . s ) by Lm2, Lm5; hence f . x = g3 . x by A5, A15, A16; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th119: :: JGRAPH_4:119 for cn being Real for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous proof let cn be Real; ::_thesis: for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds f is continuous let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) implies f is continuous ) reconsider g1 = (2 NormF) | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5; set a = cn; set b = 1 + cn; reconsider g2 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm2; assume that A1: - 1 < cn and A2: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) and A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: f is continuous A4: 1 + cn > 0 by A1, XREAL_1:148; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q <> 0. (TOP-REAL 2) by A3; then for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0 by Lm6; then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that A5: for q being Point of ((TOP-REAL 2) | K1) for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds g3 . q = r2 * (- (sqrt (abs (1 - ((((r1 / r2) - cn) / (1 + cn)) ^2))))) and A6: g3 is continuous by A4, Th9; A7: dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; then A8: dom f = dom g3 by FUNCT_2:def_1; for x being set st x in dom f holds f . x = g3 . x proof let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x ) assume A9: x in dom f ; ::_thesis: f . x = g3 . x then x in K1 by A7, A8, PRE_TOPC:8; then reconsider r = x as Point of (TOP-REAL 2) ; reconsider s = x as Point of ((TOP-REAL 2) | K1) by A9; A10: (1 + cn) ^2 > 0 by A4, SQUARE_1:12; A11: |.r.| <> 0 by A3, A9, TOPRNS_1:24; |.r.| ^2 = ((r `1) ^2) + ((r `2) ^2) by JGRAPH_3:1; then A12: ((r `1) - |.r.|) * ((r `1) + |.r.|) = - ((r `2) ^2) ; (r `2) ^2 >= 0 by XREAL_1:63; then - |.r.| <= r `1 by A12, XREAL_1:93; then (r `1) / |.r.| >= (- |.r.|) / |.r.| by XREAL_1:72; then (r `1) / |.r.| >= - 1 by A11, XCMPLX_1:197; then ((r `1) / |.r.|) - cn >= (- 1) - cn by XREAL_1:9; then A13: ((r `1) / |.r.|) - cn >= - (1 + cn) ; cn - ((r `1) / |.r.|) >= 0 by A3, A9, XREAL_1:48; then - (cn - ((r `1) / |.r.|)) <= - 0 ; then (((r `1) / |.r.|) - cn) ^2 <= (1 + cn) ^2 by A4, A13, SQUARE_1:49; then ((((r `1) / |.r.|) - cn) ^2) / ((1 + cn) ^2) <= ((1 + cn) ^2) / ((1 + cn) ^2) by A4, XREAL_1:72; then ((((r `1) / |.r.|) - cn) ^2) / ((1 + cn) ^2) <= 1 by A10, XCMPLX_1:60; then ((((r `1) / |.r.|) - cn) / (1 + cn)) ^2 <= 1 by XCMPLX_1:76; then 1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then abs (1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2)) = 1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2) by ABSVALUE:def_1; then A14: f . r = |.r.| * (- (sqrt (abs (1 - (((((r `1) / |.r.|) - cn) / (1 + cn)) ^2))))) by A2, A9; A15: ( proj1 . r = r `1 & (2 NormF) . r = |.r.| ) by Def1, PSCOMP_1:def_5; ( g2 . s = proj1 . s & g1 . s = (2 NormF) . s ) by Lm2, Lm5; hence f . x = g3 . x by A5, A14, A15; ::_thesis: verum end; hence f is continuous by A6, A8, FUNCT_1:2; ::_thesis: verum end; theorem Th120: :: JGRAPH_4:120 for cn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set sn = - (sqrt (1 - (cn ^2))); set p0 = |[cn,(- (sqrt (1 - (cn ^2))))]|; A1: |[cn,(- (sqrt (1 - (cn ^2))))]| `2 = - (sqrt (1 - (cn ^2))) by EUCLID:52; |[cn,(- (sqrt (1 - (cn ^2))))]| `1 = cn by EUCLID:52; then A2: |.|[cn,(- (sqrt (1 - (cn ^2))))]|.| = sqrt (((- (sqrt (1 - (cn ^2)))) ^2) + (cn ^2)) by A1, JGRAPH_3:1; assume A3: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then cn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (cn ^2) > 0 by XREAL_1:50; then A5: - (- (sqrt (1 - (cn ^2)))) > 0 by SQUARE_1:25; A6: now__::_thesis:_not_|[cn,(-_(sqrt_(1_-_(cn_^2))))]|_=_0._(TOP-REAL_2) assume |[cn,(- (sqrt (1 - (cn ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then - (- (- (sqrt (1 - (cn ^2))))) = - 0 by EUCLID:52, JGRAPH_2:3; hence contradiction by A4, SQUARE_1:25; ::_thesis: verum end; (- (- (sqrt (1 - (cn ^2))))) ^2 = 1 - (cn ^2) by A4, SQUARE_1:def_2; then (|[cn,(- (sqrt (1 - (cn ^2))))]| `1) / |.|[cn,(- (sqrt (1 - (cn ^2))))]|.| = cn by A2, EUCLID:52, SQUARE_1:18; then A7: |[cn,(- (sqrt (1 - (cn ^2))))]| in K0 by A3, A1, A6, A5; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A8: rng (proj2 * ((cn -FanMorphS) | K1)) c= the carrier of R^1 by TOPMETR:17; A9: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & (p8 `1) / |.p8.| >= cn & p8 `2 <= 0 & p8 <> 0. (TOP-REAL 2) ) by A3; hence x in B0 by A3; ::_thesis: verum end; A10: dom ((cn -FanMorphS) | K1) c= dom (proj1 * ((cn -FanMorphS) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((cn -FanMorphS) | K1) or x in dom (proj1 * ((cn -FanMorphS) | K1)) ) assume A11: x in dom ((cn -FanMorphS) | K1) ; ::_thesis: x in dom (proj1 * ((cn -FanMorphS) | K1)) then x in (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61; then x in dom (cn -FanMorphS) by XBOOLE_0:def_4; then A12: ( dom proj1 = the carrier of (TOP-REAL 2) & (cn -FanMorphS) . x in rng (cn -FanMorphS) ) by FUNCT_1:3, FUNCT_2:def_1; ((cn -FanMorphS) | K1) . x = (cn -FanMorphS) . x by A11, FUNCT_1:47; hence x in dom (proj1 * ((cn -FanMorphS) | K1)) by A11, A12, FUNCT_1:11; ::_thesis: verum end; A13: rng (proj1 * ((cn -FanMorphS) | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1 * ((cn -FanMorphS) | K1)) c= dom ((cn -FanMorphS) | K1) by RELAT_1:25; then dom (proj1 * ((cn -FanMorphS) | K1)) = dom ((cn -FanMorphS) | K1) by A10, XBOOLE_0:def_10 .= (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g2 = proj1 * ((cn -FanMorphS) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A13, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) A14: dom ((cn -FanMorphS) | K1) = (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A15: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A16: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A15; then A17: (cn -FanMorphS) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))))]| by A3, Th115; ((cn -FanMorphS) | K1) . p = (cn -FanMorphS) . p by A16, A15, FUNCT_1:49; then g2 . p = proj1 . |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))))]| by A16, A14, A15, A17, FUNCT_1:13 .= |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))))]| `1 by PSCOMP_1:def_5 .= |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) by EUCLID:52 ; hence g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A18: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ; A19: dom ((cn -FanMorphS) | K1) c= dom (proj2 * ((cn -FanMorphS) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((cn -FanMorphS) | K1) or x in dom (proj2 * ((cn -FanMorphS) | K1)) ) assume A20: x in dom ((cn -FanMorphS) | K1) ; ::_thesis: x in dom (proj2 * ((cn -FanMorphS) | K1)) then x in (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61; then x in dom (cn -FanMorphS) by XBOOLE_0:def_4; then A21: ( dom proj2 = the carrier of (TOP-REAL 2) & (cn -FanMorphS) . x in rng (cn -FanMorphS) ) by FUNCT_1:3, FUNCT_2:def_1; ((cn -FanMorphS) | K1) . x = (cn -FanMorphS) . x by A20, FUNCT_1:47; hence x in dom (proj2 * ((cn -FanMorphS) | K1)) by A20, A21, FUNCT_1:11; ::_thesis: verum end; dom (proj2 * ((cn -FanMorphS) | K1)) c= dom ((cn -FanMorphS) | K1) by RELAT_1:25; then dom (proj2 * ((cn -FanMorphS) | K1)) = dom ((cn -FanMorphS) | K1) by A19, XBOOLE_0:def_10 .= (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g1 = proj2 * ((cn -FanMorphS) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A8, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) ) A22: dom ((cn -FanMorphS) | K1) = (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A23: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A24: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A23; then A25: (cn -FanMorphS) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))))]| by A3, Th115; ((cn -FanMorphS) | K1) . p = (cn -FanMorphS) . p by A24, A23, FUNCT_1:49; then g1 . p = proj2 . |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))))]| by A24, A22, A23, A25, FUNCT_1:13 .= |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))))]| `2 by PSCOMP_1:def_6 .= |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52 ; hence g1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A26: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) ; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) A27: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A27; hence ( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then A28: f1 is continuous by A3, A26, Th118; A29: for x, y, s, r being real number st |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| holds f . |[x,y]| = |[s,r]| proof let x, y, s, r be real number ; ::_thesis: ( |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| implies f . |[x,y]| = |[s,r]| ) assume that A30: |[x,y]| in K1 and A31: ( s = f2 . |[x,y]| & r = f1 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[s,r]| set p99 = |[x,y]|; A32: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A30; A33: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then A34: f1 . |[x,y]| = |.|[x,y]|.| * (- (sqrt (1 - (((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 - cn)) ^2)))) by A26, A30; ((cn -FanMorphS) | K0) . |[x,y]| = (cn -FanMorphS) . |[x,y]| by A30, FUNCT_1:49 .= |[(|.|[x,y]|.| * ((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 - cn))),(|.|[x,y]|.| * (- (sqrt (1 - (((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 - cn)) ^2)))))]| by A3, A32, Th115 .= |[s,r]| by A18, A30, A31, A33, A34 ; hence f . |[x,y]| = |[s,r]| by A3; ::_thesis: verum end; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) A35: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `1) / |.p3.| >= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A35; hence ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then f2 is continuous by A3, A18, Th116; hence f is continuous by A7, A9, A28, A29, JGRAPH_2:35; ::_thesis: verum end; theorem Th121: :: JGRAPH_4:121 for cn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set sn = - (sqrt (1 - (cn ^2))); set p0 = |[cn,(- (sqrt (1 - (cn ^2))))]|; A1: |[cn,(- (sqrt (1 - (cn ^2))))]| `2 = - (sqrt (1 - (cn ^2))) by EUCLID:52; |[cn,(- (sqrt (1 - (cn ^2))))]| `1 = cn by EUCLID:52; then A2: |.|[cn,(- (sqrt (1 - (cn ^2))))]|.| = sqrt (((- (sqrt (1 - (cn ^2)))) ^2) + (cn ^2)) by A1, JGRAPH_3:1; assume A3: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then cn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (cn ^2) > 0 by XREAL_1:50; then A5: - (- (sqrt (1 - (cn ^2)))) > 0 by SQUARE_1:25; A6: now__::_thesis:_not_|[cn,(-_(sqrt_(1_-_(cn_^2))))]|_=_0._(TOP-REAL_2) assume |[cn,(- (sqrt (1 - (cn ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then - (- (- (sqrt (1 - (cn ^2))))) = - 0 by EUCLID:52, JGRAPH_2:3; hence contradiction by A4, SQUARE_1:25; ::_thesis: verum end; (- (- (sqrt (1 - (cn ^2))))) ^2 = 1 - (cn ^2) by A4, SQUARE_1:def_2; then (|[cn,(- (sqrt (1 - (cn ^2))))]| `1) / |.|[cn,(- (sqrt (1 - (cn ^2))))]|.| = cn by A2, EUCLID:52, SQUARE_1:18; then A7: |[cn,(- (sqrt (1 - (cn ^2))))]| in K0 by A3, A1, A6, A5; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; A8: rng (proj2 * ((cn -FanMorphS) | K1)) c= the carrier of R^1 by TOPMETR:17; A9: K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then ex p8 being Point of (TOP-REAL 2) st ( x = p8 & (p8 `1) / |.p8.| <= cn & p8 `2 <= 0 & p8 <> 0. (TOP-REAL 2) ) by A3; hence x in B0 by A3; ::_thesis: verum end; A10: dom ((cn -FanMorphS) | K1) c= dom (proj1 * ((cn -FanMorphS) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((cn -FanMorphS) | K1) or x in dom (proj1 * ((cn -FanMorphS) | K1)) ) assume A11: x in dom ((cn -FanMorphS) | K1) ; ::_thesis: x in dom (proj1 * ((cn -FanMorphS) | K1)) then x in (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61; then x in dom (cn -FanMorphS) by XBOOLE_0:def_4; then A12: ( dom proj1 = the carrier of (TOP-REAL 2) & (cn -FanMorphS) . x in rng (cn -FanMorphS) ) by FUNCT_1:3, FUNCT_2:def_1; ((cn -FanMorphS) | K1) . x = (cn -FanMorphS) . x by A11, FUNCT_1:47; hence x in dom (proj1 * ((cn -FanMorphS) | K1)) by A11, A12, FUNCT_1:11; ::_thesis: verum end; A13: rng (proj1 * ((cn -FanMorphS) | K1)) c= the carrier of R^1 by TOPMETR:17; dom (proj1 * ((cn -FanMorphS) | K1)) c= dom ((cn -FanMorphS) | K1) by RELAT_1:25; then dom (proj1 * ((cn -FanMorphS) | K1)) = dom ((cn -FanMorphS) | K1) by A10, XBOOLE_0:def_10 .= (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g2 = proj1 * ((cn -FanMorphS) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A13, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) A14: dom ((cn -FanMorphS) | K1) = (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A15: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A16: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A15; then A17: (cn -FanMorphS) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))))]| by A3, Th115; ((cn -FanMorphS) | K1) . p = (cn -FanMorphS) . p by A16, A15, FUNCT_1:49; then g2 . p = proj1 . |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))))]| by A16, A14, A15, A17, FUNCT_1:13 .= |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))))]| `1 by PSCOMP_1:def_5 .= |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) by EUCLID:52 ; hence g2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ; ::_thesis: verum end; then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that A18: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f2 . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ; A19: dom ((cn -FanMorphS) | K1) c= dom (proj2 * ((cn -FanMorphS) | K1)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((cn -FanMorphS) | K1) or x in dom (proj2 * ((cn -FanMorphS) | K1)) ) assume A20: x in dom ((cn -FanMorphS) | K1) ; ::_thesis: x in dom (proj2 * ((cn -FanMorphS) | K1)) then x in (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61; then x in dom (cn -FanMorphS) by XBOOLE_0:def_4; then A21: ( dom proj2 = the carrier of (TOP-REAL 2) & (cn -FanMorphS) . x in rng (cn -FanMorphS) ) by FUNCT_1:3, FUNCT_2:def_1; ((cn -FanMorphS) | K1) . x = (cn -FanMorphS) . x by A20, FUNCT_1:47; hence x in dom (proj2 * ((cn -FanMorphS) | K1)) by A20, A21, FUNCT_1:11; ::_thesis: verum end; dom (proj2 * ((cn -FanMorphS) | K1)) c= dom ((cn -FanMorphS) | K1) by RELAT_1:25; then dom (proj2 * ((cn -FanMorphS) | K1)) = dom ((cn -FanMorphS) | K1) by A19, XBOOLE_0:def_10 .= (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 .= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ; then reconsider g1 = proj2 * ((cn -FanMorphS) | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A8, FUNCT_2:2; for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds g1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) proof let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) ) A22: dom ((cn -FanMorphS) | K1) = (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by XBOOLE_1:28 ; A23: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume A24: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) then ex p3 being Point of (TOP-REAL 2) st ( p = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A23; then A25: (cn -FanMorphS) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))))]| by A3, Th115; ((cn -FanMorphS) | K1) . p = (cn -FanMorphS) . p by A24, A23, FUNCT_1:49; then g1 . p = proj2 . |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))))]| by A24, A22, A23, A25, FUNCT_1:13 .= |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))))]| `2 by PSCOMP_1:def_6 .= |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) by EUCLID:52 ; hence g1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) ; ::_thesis: verum end; then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that A26: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f1 . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) ; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) A27: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A27; hence ( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then A28: f1 is continuous by A3, A26, Th119; A29: for x, y, s, r being real number st |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| holds f . |[x,y]| = |[s,r]| proof let x, y, s, r be real number ; ::_thesis: ( |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| implies f . |[x,y]| = |[s,r]| ) assume that A30: |[x,y]| in K1 and A31: ( s = f2 . |[x,y]| & r = f1 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[s,r]| set p99 = |[x,y]|; A32: ex p3 being Point of (TOP-REAL 2) st ( |[x,y]| = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A30; A33: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then A34: f1 . |[x,y]| = |.|[x,y]|.| * (- (sqrt (1 - (((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 + cn)) ^2)))) by A26, A30; ((cn -FanMorphS) | K0) . |[x,y]| = (cn -FanMorphS) . |[x,y]| by A30, FUNCT_1:49 .= |[(|.|[x,y]|.| * ((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 + cn))),(|.|[x,y]|.| * (- (sqrt (1 - (((((|[x,y]| `1) / |.|[x,y]|.|) - cn) / (1 + cn)) ^2)))))]| by A3, A32, Th115 .= |[s,r]| by A18, A30, A31, A33, A34 ; hence f . |[x,y]| = |[s,r]| by A3; ::_thesis: verum end; for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) proof let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) A35: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) then ex p3 being Point of (TOP-REAL 2) st ( q = p3 & (p3 `1) / |.p3.| <= cn & p3 `2 <= 0 & p3 <> 0. (TOP-REAL 2) ) by A3, A35; hence ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ; ::_thesis: verum end; then f2 is continuous by A3, A18, Th117; hence f is continuous by A7, A9, A28, A29, JGRAPH_2:35; ::_thesis: verum end; theorem Th122: :: JGRAPH_4:122 for cn being Real for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= cn * |.p.| & p `2 <= 0 ) } holds K03 is closed proof defpred S1[ Point of (TOP-REAL 2)] means $1 `2 <= 0 ; let sn be Real; ::_thesis: for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= sn * |.p.| & p `2 <= 0 ) } holds K03 is closed let K003 be Subset of (TOP-REAL 2); ::_thesis: ( K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= sn * |.p.| & p `2 <= 0 ) } implies K003 is closed ) assume A1: K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= sn * |.p.| & p `2 <= 0 ) } ; ::_thesis: K003 is closed reconsider KX = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); defpred S2[ Point of (TOP-REAL 2)] means $1 `1 >= sn * |.$1.|; reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A2: { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ { p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } from DOMAIN_1:sch_10(); ( K1 is closed & KX is closed ) by Lm8, JORDAN6:8; hence K003 is closed by A1, A2, TOPS_1:8; ::_thesis: verum end; theorem Th123: :: JGRAPH_4:123 for cn being Real for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= cn * |.p.| & p `2 <= 0 ) } holds K03 is closed proof defpred S1[ Point of (TOP-REAL 2)] means $1 `2 <= 0 ; let sn be Real; ::_thesis: for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= sn * |.p.| & p `2 <= 0 ) } holds K03 is closed let K003 be Subset of (TOP-REAL 2); ::_thesis: ( K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= sn * |.p.| & p `2 <= 0 ) } implies K003 is closed ) assume A1: K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= sn * |.p.| & p `2 <= 0 ) } ; ::_thesis: K003 is closed reconsider KX = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); defpred S2[ Point of (TOP-REAL 2)] means $1 `1 <= sn * |.$1.|; reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1(); A2: { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p7 where p7 is Point of (TOP-REAL 2) : S2[p7] } /\ { p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } from DOMAIN_1:sch_10(); ( K1 is closed & KX is closed ) by Lm10, JORDAN6:8; hence K003 is closed by A1, A2, TOPS_1:8; ::_thesis: verum end; theorem Th124: :: JGRAPH_4:124 for cn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set sn = - (sqrt (1 - (cn ^2))); set p0 = |[cn,(- (sqrt (1 - (cn ^2))))]|; A1: |[cn,(- (sqrt (1 - (cn ^2))))]| `2 = - (sqrt (1 - (cn ^2))) by EUCLID:52; |[cn,(- (sqrt (1 - (cn ^2))))]| `1 = cn by EUCLID:52; then A2: |.|[cn,(- (sqrt (1 - (cn ^2))))]|.| = sqrt (((- (sqrt (1 - (cn ^2)))) ^2) + (cn ^2)) by A1, JGRAPH_3:1; assume A3: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then cn ^2 < 1 ^2 by SQUARE_1:50; then A4: 1 - (cn ^2) > 0 by XREAL_1:50; then A5: - (- (sqrt (1 - (cn ^2)))) > 0 by SQUARE_1:25; A6: now__::_thesis:_not_|[cn,(-_(sqrt_(1_-_(cn_^2))))]|_=_0._(TOP-REAL_2) assume |[cn,(- (sqrt (1 - (cn ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction then - (- (- (sqrt (1 - (cn ^2))))) = - 0 by EUCLID:52, JGRAPH_2:3; hence contradiction by A4, SQUARE_1:25; ::_thesis: verum end; then |[cn,(- (sqrt (1 - (cn ^2))))]| in K0 by A3, A1, A5; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; (- (- (sqrt (1 - (cn ^2))))) ^2 = 1 - (cn ^2) by A4, SQUARE_1:def_2; then A7: (|[cn,(- (sqrt (1 - (cn ^2))))]| `1) / |.|[cn,(- (sqrt (1 - (cn ^2))))]|.| = cn by A2, EUCLID:52, SQUARE_1:18; then A8: |[cn,(- (sqrt (1 - (cn ^2))))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5; not |[cn,(- (sqrt (1 - (cn ^2))))]| in {(0. (TOP-REAL 2))} by A6, TARSKI:def_1; then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A3, XBOOLE_0:def_5; K1 c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in D ) assume A9: x in K1 ; ::_thesis: x in D then ex p6 being Point of (TOP-REAL 2) st ( p6 = x & p6 `2 <= 0 & p6 <> 0. (TOP-REAL 2) ) by A3; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in D by A3, A9, XBOOLE_0:def_5; ::_thesis: verum end; then D = K1 \/ D by XBOOLE_1:12; then A10: (TOP-REAL 2) | K1 is SubSpace of (TOP-REAL 2) | D by TOPMETR:4; A11: { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } c= K1 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) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 ) assume x in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in K1 then ex p being Point of (TOP-REAL 2) st ( p = x & (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) ; hence x in K1 by A3; ::_thesis: verum end; A12: { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } c= K1 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) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } or x in K1 ) assume x in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in K1 then ex p being Point of (TOP-REAL 2) st ( p = x & (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) ; hence x in K1 by A3; ::_thesis: verum end; then reconsider K00 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A8, PRE_TOPC:8; the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; then A13: rng (f | K00) c= D ; |[cn,(- (sqrt (1 - (cn ^2))))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5, A7; then reconsider K11 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | K1) by A11, PRE_TOPC:8; the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; then A14: rng (f | K11) c= D ; the carrier of ((TOP-REAL 2) | B0) = the carrier of ((TOP-REAL 2) | D) ; then A15: dom f = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1 .= K1 by PRE_TOPC:8 ; then dom (f | K00) = K00 by A12, RELAT_1:62 .= the carrier of (((TOP-REAL 2) | K1) | K00) by PRE_TOPC:8 ; then reconsider f1 = f | K00 as Function of (((TOP-REAL 2) | K1) | K00),((TOP-REAL 2) | D) by A13, FUNCT_2:2; dom (f | K11) = K11 by A11, A15, RELAT_1:62 .= the carrier of (((TOP-REAL 2) | K1) | K11) by PRE_TOPC:8 ; then reconsider f2 = f | K11 as Function of (((TOP-REAL 2) | K1) | K11),((TOP-REAL 2) | D) by A14, FUNCT_2:2; A16: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; defpred S1[ Point of (TOP-REAL 2)] means ( ($1 `1) / |.$1.| >= cn & $1 `2 <= 0 & $1 <> 0. (TOP-REAL 2) ); A17: dom f2 = the carrier of (((TOP-REAL 2) | K1) | K11) by FUNCT_2:def_1 .= K11 by PRE_TOPC:8 ; { p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K001 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A8; A18: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; defpred S2[ Point of (TOP-REAL 2)] means ( $1 `1 >= cn * |.$1.| & $1 `2 <= 0 ); { p where p is Point of (TOP-REAL 2) : S2[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K003 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= cn * |.p.| & p `2 <= 0 ) } as Subset of (TOP-REAL 2) ; defpred S3[ Point of (TOP-REAL 2)] means ( ($1 `1) / |.$1.| <= cn & $1 `2 <= 0 & $1 <> 0. (TOP-REAL 2) ); A19: { p where p is Point of (TOP-REAL 2) : S3[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); A20: rng ((cn -FanMorphS) | K001) c= K1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((cn -FanMorphS) | K001) or y in K1 ) assume y in rng ((cn -FanMorphS) | K001) ; ::_thesis: y in K1 then consider x being set such that A21: x in dom ((cn -FanMorphS) | K001) and A22: y = ((cn -FanMorphS) | K001) . x by FUNCT_1:def_3; x in dom (cn -FanMorphS) by A21, RELAT_1:57; then reconsider q = x as Point of (TOP-REAL 2) ; A23: y = (cn -FanMorphS) . q by A21, A22, FUNCT_1:47; dom ((cn -FanMorphS) | K001) = (dom (cn -FanMorphS)) /\ K001 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K001 by FUNCT_2:def_1 .= K001 by XBOOLE_1:28 ; then A24: ex p2 being Point of (TOP-REAL 2) st ( p2 = q & (p2 `1) / |.p2.| >= cn & p2 `2 <= 0 & p2 <> 0. (TOP-REAL 2) ) by A21; then A25: ((q `1) / |.q.|) - cn >= 0 by XREAL_1:48; |.q.| <> 0 by A24, TOPRNS_1:24; then A26: |.q.| ^2 > 0 ^2 by SQUARE_1:12; set q4 = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]|; A27: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn)) by EUCLID:52; A28: 1 - cn > 0 by A3, XREAL_1:149; 0 <= (q `2) ^2 by XREAL_1:63; then 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) by XREAL_1:7; then (q `1) ^2 <= |.q.| ^2 by JGRAPH_3:1; then ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A26, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `1) / |.q.| by SQUARE_1:51; then 1 - cn >= ((q `1) / |.q.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((q `1) / |.q.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A28, XREAL_1:72; then - 1 <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A28, XCMPLX_1:197; then ((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A28, A25, SQUARE_1:49; then A29: 1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A30: 1 - ((- ((((q `1) / |.q.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; sqrt (1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2)) >= 0 by A29, SQUARE_1:def_2; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) ^2) / ((1 - cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((q `1) / |.q.|) - cn) ^2) / ((1 - cn) ^2))) >= 0 ; then A31: sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) >= 0 by XCMPLX_1:76; A32: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52; then A33: (|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) by A30, SQUARE_1:def_2 ; |.|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]|.| ^2 = ((|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| `1) ^2) + ((|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A27, A33 ; then A34: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| <> 0. (TOP-REAL 2) by A26, TOPRNS_1:23; (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by A3, A24, Th115; hence y in K1 by A3, A23, A32, A31, A34; ::_thesis: verum end; A35: dom (cn -FanMorphS) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then dom ((cn -FanMorphS) | K001) = K001 by RELAT_1:62 .= the carrier of ((TOP-REAL 2) | K001) by PRE_TOPC:8 ; then reconsider f3 = (cn -FanMorphS) | K001 as Function of ((TOP-REAL 2) | K001),((TOP-REAL 2) | K1) by A18, A20, FUNCT_2:2; A36: K003 is closed by Th122; defpred S4[ Point of (TOP-REAL 2)] means ( $1 `1 <= cn * |.$1.| & $1 `2 <= 0 ); { p where p is Point of (TOP-REAL 2) : S4[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7(); then reconsider K004 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= cn * |.p.| & p `2 <= 0 ) } as Subset of (TOP-REAL 2) ; A37: K004 /\ K1 c= K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K004 /\ K1 or x in K11 ) assume A38: x in K004 /\ K1 ; ::_thesis: x in K11 then x in K004 by XBOOLE_0:def_4; then consider q1 being Point of (TOP-REAL 2) such that A39: q1 = x and A40: q1 `1 <= cn * |.q1.| and q1 `2 <= 0 ; x in K1 by A38, XBOOLE_0:def_4; then A41: ex q2 being Point of (TOP-REAL 2) st ( q2 = x & q2 `2 <= 0 & q2 <> 0. (TOP-REAL 2) ) by A3; (q1 `1) / |.q1.| <= (cn * |.q1.|) / |.q1.| by A40, XREAL_1:72; then (q1 `1) / |.q1.| <= cn by A39, A41, TOPRNS_1:24, XCMPLX_1:89; hence x in K11 by A39, A41; ::_thesis: verum end; A42: K004 is closed by Th123; the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then ( ((TOP-REAL 2) | K1) | K00 = (TOP-REAL 2) | K001 & f1 = f3 ) by A3, FUNCT_1:51, GOBOARD9:2; then A43: f1 is continuous by A3, A10, Th120, PRE_TOPC:26; A44: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def_5; |[cn,(- (sqrt (1 - (cn ^2))))]| in { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } by A1, A6, A5, A7; then reconsider K111 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of (TOP-REAL 2) by A19; A45: rng ((cn -FanMorphS) | K111) c= K1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((cn -FanMorphS) | K111) or y in K1 ) assume y in rng ((cn -FanMorphS) | K111) ; ::_thesis: y in K1 then consider x being set such that A46: x in dom ((cn -FanMorphS) | K111) and A47: y = ((cn -FanMorphS) | K111) . x by FUNCT_1:def_3; x in dom (cn -FanMorphS) by A46, RELAT_1:57; then reconsider q = x as Point of (TOP-REAL 2) ; A48: y = (cn -FanMorphS) . q by A46, A47, FUNCT_1:47; dom ((cn -FanMorphS) | K111) = (dom (cn -FanMorphS)) /\ K111 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K111 by FUNCT_2:def_1 .= K111 by XBOOLE_1:28 ; then A49: ex p2 being Point of (TOP-REAL 2) st ( p2 = q & (p2 `1) / |.p2.| <= cn & p2 `2 <= 0 & p2 <> 0. (TOP-REAL 2) ) by A46; then A50: ((q `1) / |.q.|) - cn <= 0 by XREAL_1:47; |.q.| <> 0 by A49, TOPRNS_1:24; then A51: |.q.| ^2 > 0 ^2 by SQUARE_1:12; set q4 = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]|; A52: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) by EUCLID:52; A53: 1 + cn > 0 by A3, XREAL_1:148; 0 <= (q `2) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A51, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `1) / |.q.| by SQUARE_1:51; then (- 1) - cn <= ((q `1) / |.q.|) - cn by XREAL_1:9; then (- (1 + cn)) / (1 + cn) <= (((q `1) / |.q.|) - cn) / (1 + cn) by A53, XREAL_1:72; then - 1 <= (((q `1) / |.q.|) - cn) / (1 + cn) by A53, XCMPLX_1:197; then A54: ((((q `1) / |.q.|) - cn) / (1 + cn)) ^2 <= 1 ^2 by A53, A50, SQUARE_1:49; then A55: 1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2) >= 0 by XREAL_1:48; 1 - ((- ((((q `1) / |.q.|) - cn) / (1 + cn))) ^2) >= 0 by A54, XREAL_1:48; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2) >= 0 by XCMPLX_1:187; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) ^2) / ((1 + cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((q `1) / |.q.|) - cn) ^2) / ((1 + cn) ^2))) >= 0 ; then A56: sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) >= 0 by XCMPLX_1:76; A57: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))) by EUCLID:52; then A58: (|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) by A55, SQUARE_1:def_2 ; |.|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]|.| ^2 = ((|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| `1) ^2) + ((|[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A52, A58 ; then A59: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| <> 0. (TOP-REAL 2) by A51, TOPRNS_1:23; (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| by A3, A49, Th115; hence y in K1 by A3, A48, A57, A56, A59; ::_thesis: verum end; dom ((cn -FanMorphS) | K111) = K111 by A35, RELAT_1:62 .= the carrier of ((TOP-REAL 2) | K111) by PRE_TOPC:8 ; then reconsider f4 = (cn -FanMorphS) | K111 as Function of ((TOP-REAL 2) | K111),((TOP-REAL 2) | K1) by A16, A45, FUNCT_2:2; the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8; then ( ((TOP-REAL 2) | K1) | K11 = (TOP-REAL 2) | K111 & f2 = f4 ) by A3, FUNCT_1:51, GOBOARD9:2; then A60: f2 is continuous by A3, A10, Th121, PRE_TOPC:26; set T1 = ((TOP-REAL 2) | K1) | K00; set T2 = ((TOP-REAL 2) | K1) | K11; A61: [#] (((TOP-REAL 2) | K1) | K11) = K11 by PRE_TOPC:def_5; K11 c= K004 /\ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K11 or x in K004 /\ K1 ) assume x in K11 ; ::_thesis: x in K004 /\ K1 then consider p being Point of (TOP-REAL 2) such that A62: p = x and A63: (p `1) / |.p.| <= cn and A64: p `2 <= 0 and A65: p <> 0. (TOP-REAL 2) ; ((p `1) / |.p.|) * |.p.| <= cn * |.p.| by A63, XREAL_1:64; then p `1 <= cn * |.p.| by A65, TOPRNS_1:24, XCMPLX_1:87; then A66: x in K004 by A62, A64; x in K1 by A3, A62, A64, A65; hence x in K004 /\ K1 by A66, XBOOLE_0:def_4; ::_thesis: verum end; then K11 = K004 /\ ([#] ((TOP-REAL 2) | K1)) by A44, A37, XBOOLE_0:def_10; then A67: K11 is closed by A42, PRE_TOPC:13; A68: K003 /\ K1 c= K00 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K003 /\ K1 or x in K00 ) assume A69: x in K003 /\ K1 ; ::_thesis: x in K00 then x in K003 by XBOOLE_0:def_4; then consider q1 being Point of (TOP-REAL 2) such that A70: q1 = x and A71: q1 `1 >= cn * |.q1.| and q1 `2 <= 0 ; x in K1 by A69, XBOOLE_0:def_4; then A72: ex q2 being Point of (TOP-REAL 2) st ( q2 = x & q2 `2 <= 0 & q2 <> 0. (TOP-REAL 2) ) by A3; (q1 `1) / |.q1.| >= (cn * |.q1.|) / |.q1.| by A71, XREAL_1:72; then (q1 `1) / |.q1.| >= cn by A70, A72, TOPRNS_1:24, XCMPLX_1:89; hence x in K00 by A70, A72; ::_thesis: verum end; A73: the carrier of ((TOP-REAL 2) | K1) = K0 by PRE_TOPC:8; A74: D <> {} ; A75: [#] (((TOP-REAL 2) | K1) | K00) = K00 by PRE_TOPC:def_5; A76: for p being set st p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) holds f1 . p = f2 . p proof let p be set ; ::_thesis: ( p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) implies f1 . p = f2 . p ) assume A77: p in ([#] (((TOP-REAL 2) | K1) | K00)) /\ ([#] (((TOP-REAL 2) | K1) | K11)) ; ::_thesis: f1 . p = f2 . p then p in K00 by A75, XBOOLE_0:def_4; hence f1 . p = f . p by FUNCT_1:49 .= f2 . p by A61, A77, FUNCT_1:49 ; ::_thesis: verum end; K00 c= K003 /\ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K00 or x in K003 /\ K1 ) assume x in K00 ; ::_thesis: x in K003 /\ K1 then consider p being Point of (TOP-REAL 2) such that A78: p = x and A79: (p `1) / |.p.| >= cn and A80: p `2 <= 0 and A81: p <> 0. (TOP-REAL 2) ; ((p `1) / |.p.|) * |.p.| >= cn * |.p.| by A79, XREAL_1:64; then p `1 >= cn * |.p.| by A81, TOPRNS_1:24, XCMPLX_1:87; then A82: x in K003 by A78, A80; x in K1 by A3, A78, A80, A81; hence x in K003 /\ K1 by A82, XBOOLE_0:def_4; ::_thesis: verum end; then K00 = K003 /\ ([#] ((TOP-REAL 2) | K1)) by A44, A68, XBOOLE_0:def_10; then A83: K00 is closed by A36, PRE_TOPC:13; A84: K1 c= K00 \/ K11 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in K00 \/ K11 ) assume x in K1 ; ::_thesis: x in K00 \/ K11 then consider p being Point of (TOP-REAL 2) such that A85: ( p = x & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) by A3; percases ( (p `1) / |.p.| >= cn or (p `1) / |.p.| < cn ) ; suppose (p `1) / |.p.| >= cn ; ::_thesis: x in K00 \/ K11 then x in K00 by A85; hence x in K00 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; suppose (p `1) / |.p.| < cn ; ::_thesis: x in K00 \/ K11 then x in K11 by A85; hence x in K00 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum end; end; end; then ([#] (((TOP-REAL 2) | K1) | K00)) \/ ([#] (((TOP-REAL 2) | K1) | K11)) = [#] ((TOP-REAL 2) | K1) by A75, A61, A44, XBOOLE_0:def_10; then consider h being Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) such that A86: h = f1 +* f2 and A87: h is continuous by A75, A61, A83, A67, A43, A60, A76, JGRAPH_2:1; A88: dom h = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1; A89: dom f1 = the carrier of (((TOP-REAL 2) | K1) | K00) by FUNCT_2:def_1 .= K00 by PRE_TOPC:8 ; A90: for y being set st y in dom h holds h . y = f . y proof let y be set ; ::_thesis: ( y in dom h implies h . y = f . y ) assume A91: y in dom h ; ::_thesis: h . y = f . y percases ( ( y in K00 & not y in K11 ) or y in K11 ) by A84, A88, A73, A91, XBOOLE_0:def_3; supposeA92: ( y in K00 & not y in K11 ) ; ::_thesis: h . y = f . y then y in (dom f1) \/ (dom f2) by A89, XBOOLE_0:def_3; hence h . y = f1 . y by A17, A86, A92, FUNCT_4:def_1 .= f . y by A92, FUNCT_1:49 ; ::_thesis: verum end; supposeA93: y in K11 ; ::_thesis: h . y = f . y then y in (dom f1) \/ (dom f2) by A17, XBOOLE_0:def_3; hence h . y = f2 . y by A17, A86, A93, FUNCT_4:def_1 .= f . y by A93, FUNCT_1:49 ; ::_thesis: verum end; end; end; K0 = the carrier of ((TOP-REAL 2) | K0) by PRE_TOPC:8 .= dom f by A74, FUNCT_2:def_1 ; hence f is continuous by A87, A88, A90, FUNCT_1:2, PRE_TOPC:8; ::_thesis: verum end; theorem Th125: :: JGRAPH_4:125 for cn being Real for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) set sn = sqrt (1 - (cn ^2)); set p0 = |[cn,(sqrt (1 - (cn ^2)))]|; A1: |[cn,(sqrt (1 - (cn ^2)))]| `2 = sqrt (1 - (cn ^2)) by EUCLID:52; assume A2: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous then cn ^2 < 1 ^2 by SQUARE_1:50; then A3: 1 - (cn ^2) > 0 by XREAL_1:50; then sqrt (1 - (cn ^2)) > 0 by SQUARE_1:25; then |[cn,(sqrt (1 - (cn ^2)))]| in K0 by A2, A1, JGRAPH_2:3; then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ; |[cn,(sqrt (1 - (cn ^2)))]| `2 > 0 by A1, A3, SQUARE_1:25; then not |[cn,(sqrt (1 - (cn ^2)))]| in {(0. (TOP-REAL 2))} by JGRAPH_2:3, TARSKI:def_1; then reconsider D = B0 as non empty Subset of (TOP-REAL 2) by A2, XBOOLE_0:def_5; A4: K1 c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in D ) assume x in K1 ; ::_thesis: x in D then consider p2 being Point of (TOP-REAL 2) such that A5: p2 = x and p2 `2 >= 0 and A6: p2 <> 0. (TOP-REAL 2) by A2; not p2 in {(0. (TOP-REAL 2))} by A6, TARSKI:def_1; hence x in D by A2, A5, XBOOLE_0:def_5; ::_thesis: verum end; for p being Point of ((TOP-REAL 2) | K1) for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) proof let p be Point of ((TOP-REAL 2) | K1); ::_thesis: for V being Subset of ((TOP-REAL 2) | D) st f . p in V & V is open holds ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) let V be Subset of ((TOP-REAL 2) | D); ::_thesis: ( f . p in V & V is open implies ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) ) assume that A7: f . p in V and A8: V is open ; ::_thesis: ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) consider V2 being Subset of (TOP-REAL 2) such that A9: V2 is open and A10: V2 /\ ([#] ((TOP-REAL 2) | D)) = V by A8, TOPS_2:24; reconsider W2 = V2 /\ ([#] ((TOP-REAL 2) | K1)) as Subset of ((TOP-REAL 2) | K1) ; A11: [#] ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:def_5; then A12: f . p = (cn -FanMorphS) . p by A2, FUNCT_1:49; A13: f .: W2 c= V proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f .: W2 or y in V ) assume y in f .: W2 ; ::_thesis: y in V then consider x being set such that A14: x in dom f and A15: x in W2 and A16: y = f . x by FUNCT_1:def_6; f is Function of ((TOP-REAL 2) | K1),((TOP-REAL 2) | D) ; then dom f = K1 by A11, FUNCT_2:def_1; then consider p4 being Point of (TOP-REAL 2) such that A17: x = p4 and A18: p4 `2 >= 0 and p4 <> 0. (TOP-REAL 2) by A2, A14; A19: p4 in V2 by A15, A17, XBOOLE_0:def_4; p4 in [#] ((TOP-REAL 2) | K1) by A14, A17; then p4 in D by A4, A11; then A20: p4 in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; f . p4 = (cn -FanMorphS) . p4 by A2, A11, A14, A17, FUNCT_1:49 .= p4 by A18, Th113 ; hence y in V by A10, A16, A17, A19, A20, XBOOLE_0:def_4; ::_thesis: verum end; p in the carrier of ((TOP-REAL 2) | K1) ; then consider q being Point of (TOP-REAL 2) such that A21: q = p and A22: q `2 >= 0 and q <> 0. (TOP-REAL 2) by A2, A11; (cn -FanMorphS) . q = q by A22, Th113; then p in V2 by A7, A10, A12, A21, XBOOLE_0:def_4; then A23: p in W2 by XBOOLE_0:def_4; W2 is open by A9, TOPS_2:24; hence ex W being Subset of ((TOP-REAL 2) | K1) st ( p in W & W is open & f .: W c= V ) by A23, A13; ::_thesis: verum end; hence f is continuous by JGRAPH_2:10; ::_thesis: verum end; theorem Th126: :: JGRAPH_4:126 for cn being Real for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:8; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & p8 `2 <= 0 & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; hence f is continuous by A1, Th124; ::_thesis: verum end; theorem Th127: :: JGRAPH_4:127 for cn being Real for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous proof let cn be Real; ::_thesis: for B0 being Subset of (TOP-REAL 2) for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0) for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds f is continuous let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies f is continuous ) the carrier of ((TOP-REAL 2) | B0) = B0 by PRE_TOPC:8; then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1; assume A1: ( - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous K0 c= B0 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 ) assume x in K0 ; ::_thesis: x in B0 then A2: ex p8 being Point of (TOP-REAL 2) st ( x = p8 & p8 `2 >= 0 & p8 <> 0. (TOP-REAL 2) ) by A1; then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1; hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum end; then ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7; hence f is continuous by A1, Th125; ::_thesis: verum end; theorem Th128: :: JGRAPH_4:128 for cn being Real for p being Point of (TOP-REAL 2) holds |.((cn -FanMorphS) . p).| = |.p.| proof let cn be Real; ::_thesis: for p being Point of (TOP-REAL 2) holds |.((cn -FanMorphS) . p).| = |.p.| let p be Point of (TOP-REAL 2); ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| set f = cn -FanMorphS ; set z = (cn -FanMorphS) . p; set q = p; reconsider qz = (cn -FanMorphS) . p as Point of (TOP-REAL 2) ; percases ( ( (p `1) / |.p.| >= cn & p `2 < 0 ) or ( (p `1) / |.p.| < cn & p `2 < 0 ) or p `2 >= 0 ) ; supposeA1: ( (p `1) / |.p.| >= cn & p `2 < 0 ) ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| then A2: (cn -FanMorphS) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))))]| by Th113; then A3: qz `2 = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52; A4: qz `1 = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) by A2, EUCLID:52; A5: ((p `1) / |.p.|) - cn >= 0 by A1, XREAL_1:48; A6: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; |.p.| <> 0 by A1, JGRAPH_2:3, TOPRNS_1:24; then A7: |.p.| ^2 > 0 by SQUARE_1:12; 0 <= (p `2) ^2 by XREAL_1:63; then 0 + ((p `1) ^2) <= ((p `1) ^2) + ((p `2) ^2) by XREAL_1:7; then ((p `1) ^2) / (|.p.| ^2) <= (|.p.| ^2) / (|.p.| ^2) by A6, XREAL_1:72; then ((p `1) ^2) / (|.p.| ^2) <= 1 by A7, XCMPLX_1:60; then ((p `1) / |.p.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p `1) / |.p.| by SQUARE_1:51; then A8: 1 - cn >= ((p `1) / |.p.|) - cn by XREAL_1:9; percases ( 1 - cn = 0 or 1 - cn <> 0 ) ; supposeA9: 1 - cn = 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| A10: (((p `1) / |.p.|) - cn) / (1 - cn) = (((p `1) / |.p.|) - cn) * ((1 - cn) ") by XCMPLX_0:def_9 .= (((p `1) / |.p.|) - cn) * 0 by A9 .= 0 ; then 1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2) = 1 ; then (cn -FanMorphS) . p = |[(|.p.| * 0),(|.p.| * (- 1))]| by A1, A10, Th113, SQUARE_1:18 .= |[0,(- |.p.|)]| ; then ( ((cn -FanMorphS) . p) `2 = - |.p.| & ((cn -FanMorphS) . p) `1 = 0 ) by EUCLID:52; then |.((cn -FanMorphS) . p).| = sqrt (((- |.p.|) ^2) + (0 ^2)) by JGRAPH_3:1 .= sqrt (|.p.| ^2) .= |.p.| by SQUARE_1:22 ; hence |.((cn -FanMorphS) . p).| = |.p.| ; ::_thesis: verum end; supposeA11: 1 - cn <> 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| percases ( 1 - cn > 0 or 1 - cn < 0 ) by A11; supposeA12: 1 - cn > 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| - (1 - cn) <= - (((p `1) / |.p.|) - cn) by A8, XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p `1) / |.p.|) - cn)) / (1 - cn) by A12, XREAL_1:72; then - 1 <= (- (((p `1) / |.p.|) - cn)) / (1 - cn) by A12, XCMPLX_1:197; then ((- (((p `1) / |.p.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A5, A12, SQUARE_1:49; then 1 - (((- (((p `1) / |.p.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A13: 1 - ((- ((((p `1) / |.p.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; A14: (qz `2) ^2 = (|.p.| ^2) * ((sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ^2) by A3 .= (|.p.| ^2) * (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)) by A13, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.p.| ^2 by A4, A14 ; then sqrt (|.qz.| ^2) = |.p.| by SQUARE_1:22; hence |.((cn -FanMorphS) . p).| = |.p.| by SQUARE_1:22; ::_thesis: verum end; supposeA15: 1 - cn < 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| 0 + ((p `1) ^2) < ((p `1) ^2) + ((p `2) ^2) by A1, SQUARE_1:12, XREAL_1:8; then ((p `1) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A7, A6, XREAL_1:74; then ((p `1) ^2) / (|.p.| ^2) < 1 by A7, XCMPLX_1:60; then ((p `1) / |.p.|) ^2 < 1 by XCMPLX_1:76; then A16: 1 > (p `1) / |.p.| by SQUARE_1:52; ((p `1) / |.p.|) - cn >= 0 by A1, XREAL_1:48; hence |.((cn -FanMorphS) . p).| = |.p.| by A15, A16, XREAL_1:9; ::_thesis: verum end; end; end; end; end; supposeA17: ( (p `1) / |.p.| < cn & p `2 < 0 ) ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| then |.p.| <> 0 by JGRAPH_2:3, TOPRNS_1:24; then A18: |.p.| ^2 > 0 by SQUARE_1:12; A19: ((p `1) / |.p.|) - cn < 0 by A17, XREAL_1:49; A20: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; 0 <= (p `2) ^2 by XREAL_1:63; then 0 + ((p `1) ^2) <= ((p `1) ^2) + ((p `2) ^2) by XREAL_1:7; then ((p `1) ^2) / (|.p.| ^2) <= (|.p.| ^2) / (|.p.| ^2) by A20, XREAL_1:72; then ((p `1) ^2) / (|.p.| ^2) <= 1 by A18, XCMPLX_1:60; then ((p `1) / |.p.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p `1) / |.p.| by SQUARE_1:51; then A21: (- 1) - cn <= ((p `1) / |.p.|) - cn by XREAL_1:9; A22: (cn -FanMorphS) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))))]| by A17, Th114; then A23: qz `2 = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) by EUCLID:52; A24: qz `1 = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) by A22, EUCLID:52; percases ( 1 + cn = 0 or 1 + cn <> 0 ) ; supposeA25: 1 + cn = 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| (((p `1) / |.p.|) - cn) / (1 + cn) = (((p `1) / |.p.|) - cn) * ((1 + cn) ") by XCMPLX_0:def_9 .= (((p `1) / |.p.|) - cn) * 0 by A25 .= 0 ; then ( ((cn -FanMorphS) . p) `2 = - |.p.| & ((cn -FanMorphS) . p) `1 = 0 ) by A22, EUCLID:52, SQUARE_1:18; then |.((cn -FanMorphS) . p).| = sqrt (((- |.p.|) ^2) + (0 ^2)) by JGRAPH_3:1 .= sqrt (|.p.| ^2) .= |.p.| by SQUARE_1:22 ; hence |.((cn -FanMorphS) . p).| = |.p.| ; ::_thesis: verum end; supposeA26: 1 + cn <> 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| percases ( 1 + cn > 0 or 1 + cn < 0 ) by A26; supposeA27: 1 + cn > 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| then (- (1 + cn)) / (1 + cn) <= (((p `1) / |.p.|) - cn) / (1 + cn) by A21, XREAL_1:72; then - 1 <= (((p `1) / |.p.|) - cn) / (1 + cn) by A27, XCMPLX_1:197; then ((((p `1) / |.p.|) - cn) / (1 + cn)) ^2 <= 1 ^2 by A19, A27, SQUARE_1:49; then A28: 1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2) >= 0 by XREAL_1:48; A29: (qz `2) ^2 = (|.p.| ^2) * ((sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ^2) by A23 .= (|.p.| ^2) * (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)) by A28, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.p.| ^2 by A24, A29 ; then sqrt (|.qz.| ^2) = |.p.| by SQUARE_1:22; hence |.((cn -FanMorphS) . p).| = |.p.| by SQUARE_1:22; ::_thesis: verum end; supposeA30: 1 + cn < 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| 0 + ((p `1) ^2) < ((p `1) ^2) + ((p `2) ^2) by A17, SQUARE_1:12, XREAL_1:8; then ((p `1) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A18, A20, XREAL_1:74; then ((p `1) ^2) / (|.p.| ^2) < 1 by A18, XCMPLX_1:60; then ((p `1) / |.p.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (p `1) / |.p.| by SQUARE_1:52; then A31: ((p `1) / |.p.|) - cn > (- 1) - cn by XREAL_1:9; - (1 + cn) > - 0 by A30, XREAL_1:24; hence |.((cn -FanMorphS) . p).| = |.p.| by A17, A31, XREAL_1:49; ::_thesis: verum end; end; end; end; end; suppose p `2 >= 0 ; ::_thesis: |.((cn -FanMorphS) . p).| = |.p.| hence |.((cn -FanMorphS) . p).| = |.p.| by Th113; ::_thesis: verum end; end; end; theorem Th129: :: JGRAPH_4:129 for cn being Real for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds (cn -FanMorphS) . x in K0 proof let cn be Real; ::_thesis: for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds (cn -FanMorphS) . x in K0 let x, K0 be set ; ::_thesis: ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } implies (cn -FanMorphS) . x in K0 ) assume A1: ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: (cn -FanMorphS) . x in K0 then consider p being Point of (TOP-REAL 2) such that A2: p = x and A3: p `2 <= 0 and A4: p <> 0. (TOP-REAL 2) ; A5: now__::_thesis:_not_|.p.|_<=_0 assume |.p.| <= 0 ; ::_thesis: contradiction then |.p.| = 0 ; hence contradiction by A4, TOPRNS_1:24; ::_thesis: verum end; then A6: |.p.| ^2 > 0 by SQUARE_1:12; percases ( (p `1) / |.p.| <= cn or (p `1) / |.p.| > cn ) ; supposeA7: (p `1) / |.p.| <= cn ; ::_thesis: (cn -FanMorphS) . x in K0 reconsider p9 = (cn -FanMorphS) . p as Point of (TOP-REAL 2) ; (cn -FanMorphS) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))))]| by A1, A3, A4, A7, Th115; then A8: p9 `2 = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) by EUCLID:52; A9: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; A10: 1 + cn > 0 by A1, XREAL_1:148; percases ( p `2 = 0 or p `2 <> 0 ) ; suppose p `2 = 0 ; ::_thesis: (cn -FanMorphS) . x in K0 hence (cn -FanMorphS) . x in K0 by A1, A2, Th113; ::_thesis: verum end; suppose p `2 <> 0 ; ::_thesis: (cn -FanMorphS) . x in K0 then 0 + ((p `1) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8; then ((p `1) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A9, XREAL_1:74; then ((p `1) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60; then ((p `1) / |.p.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (p `1) / |.p.| by SQUARE_1:52; then (- 1) - cn < ((p `1) / |.p.|) - cn by XREAL_1:9; then ((- 1) * (1 + cn)) / (1 + cn) < (((p `1) / |.p.|) - cn) / (1 + cn) by A10, XREAL_1:74; then A11: - 1 < (((p `1) / |.p.|) - cn) / (1 + cn) by A10, XCMPLX_1:89; ((p `1) / |.p.|) - cn <= 0 by A7, XREAL_1:47; then 1 ^2 > ((((p `1) / |.p.|) - cn) / (1 + cn)) ^2 by A10, A11, SQUARE_1:50; then 1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2) > 0 by XREAL_1:50; then - (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) > 0 by SQUARE_1:25; then - (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) < 0 ; then |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) < 0 by A5, XREAL_1:132; hence (cn -FanMorphS) . x in K0 by A1, A2, A8, JGRAPH_2:3; ::_thesis: verum end; end; end; supposeA12: (p `1) / |.p.| > cn ; ::_thesis: (cn -FanMorphS) . x in K0 reconsider p9 = (cn -FanMorphS) . p as Point of (TOP-REAL 2) ; (cn -FanMorphS) . p = |[(|.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn))),(|.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))))]| by A1, A3, A4, A12, Th115; then A13: p9 `2 = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52; A14: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1; A15: 1 - cn > 0 by A1, XREAL_1:149; percases ( p `2 = 0 or p `2 <> 0 ) ; suppose p `2 = 0 ; ::_thesis: (cn -FanMorphS) . x in K0 hence (cn -FanMorphS) . x in K0 by A1, A2, Th113; ::_thesis: verum end; suppose p `2 <> 0 ; ::_thesis: (cn -FanMorphS) . x in K0 then 0 + ((p `1) ^2) < ((p `1) ^2) + ((p `2) ^2) by SQUARE_1:12, XREAL_1:8; then ((p `1) ^2) / (|.p.| ^2) < (|.p.| ^2) / (|.p.| ^2) by A6, A14, XREAL_1:74; then ((p `1) ^2) / (|.p.| ^2) < 1 by A6, XCMPLX_1:60; then ((p `1) / |.p.|) ^2 < 1 by XCMPLX_1:76; then (p `1) / |.p.| < 1 by SQUARE_1:52; then ((p `1) / |.p.|) - cn < 1 - cn by XREAL_1:9; then (((p `1) / |.p.|) - cn) / (1 - cn) < (1 - cn) / (1 - cn) by A15, XREAL_1:74; then A16: (((p `1) / |.p.|) - cn) / (1 - cn) < 1 by A15, XCMPLX_1:60; ( - (1 - cn) < - 0 & ((p `1) / |.p.|) - cn >= cn - cn ) by A12, A15, XREAL_1:9, XREAL_1:24; then ((- 1) * (1 - cn)) / (1 - cn) < (((p `1) / |.p.|) - cn) / (1 - cn) by A15, XREAL_1:74; then - 1 < (((p `1) / |.p.|) - cn) / (1 - cn) by A15, XCMPLX_1:89; then 1 ^2 > ((((p `1) / |.p.|) - cn) / (1 - cn)) ^2 by A16, SQUARE_1:50; then 1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2) > 0 by XREAL_1:50; then - (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) > 0 by SQUARE_1:25; then - (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) < 0 ; then p9 `2 < 0 by A5, A13, XREAL_1:132; hence (cn -FanMorphS) . x in K0 by A1, A2, JGRAPH_2:3; ::_thesis: verum end; end; end; end; end; theorem Th130: :: JGRAPH_4:130 for cn being Real for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds (cn -FanMorphS) . x in K0 proof let cn be Real; ::_thesis: for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds (cn -FanMorphS) . x in K0 let x, K0 be set ; ::_thesis: ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } implies (cn -FanMorphS) . x in K0 ) assume A1: ( - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: (cn -FanMorphS) . x in K0 then ex p being Point of (TOP-REAL 2) st ( p = x & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) ; hence (cn -FanMorphS) . x in K0 by A1, Th113; ::_thesis: verum end; theorem Th131: :: JGRAPH_4:131 for cn being Real for D being non empty Subset of (TOP-REAL 2) st - 1 < cn & cn < 1 & D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphS) | D & h is continuous ) proof set Y1 = |[0,1]|; defpred S1[ Point of (TOP-REAL 2)] means $1 `2 <= 0 ; reconsider B0 = {(0. (TOP-REAL 2))} as Subset of (TOP-REAL 2) ; let cn be Real; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st - 1 < cn & cn < 1 & D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphS) | D & h is continuous ) let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & D ` = {(0. (TOP-REAL 2))} implies ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphS) | D & h is continuous ) ) assume that A1: ( - 1 < cn & cn < 1 ) and A2: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphS) | D & h is continuous ) A3: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; dom (cn -FanMorphS) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; then A4: dom ((cn -FanMorphS) | D) = the carrier of (TOP-REAL 2) /\ D by RELAT_1:61 .= the carrier of ((TOP-REAL 2) | D) by A3, XBOOLE_1:28 ; ( |[0,(- 1)]| `2 = - 1 & |[0,(- 1)]| <> 0. (TOP-REAL 2) ) by EUCLID:52, JGRAPH_2:3; then A5: |[0,(- 1)]| in { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } ; |[0,1]| `2 = 1 by EUCLID:52; then A6: |[0,1]| in { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } by JGRAPH_2:3; A7: D = B0 ` by A2 .= NonZero (TOP-REAL 2) by SUBSET_1:def_4 ; { p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) from JGRAPH_4:sch_1(A7); then reconsider K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A5; A8: K0 = the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; defpred S2[ Point of (TOP-REAL 2)] means $1 `2 >= 0 ; { p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D) from JGRAPH_4:sch_1(A7); then reconsider K1 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A6; A9: ( K0 is closed & K1 is closed ) by A7, Th62, Th63; A10: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8; A11: rng ((cn -FanMorphS) | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((cn -FanMorphS) | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) ) assume y in rng ((cn -FanMorphS) | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0) then consider x being set such that A12: x in dom ((cn -FanMorphS) | K0) and A13: y = ((cn -FanMorphS) | K0) . x by FUNCT_1:def_3; x in (dom (cn -FanMorphS)) /\ K0 by A12, RELAT_1:61; then A14: x in K0 by XBOOLE_0:def_4; K0 c= the carrier of (TOP-REAL 2) by A10, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A14; (cn -FanMorphS) . p = y by A13, A14, FUNCT_1:49; then y in K0 by A1, A14, Th129; hence y in the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; ::_thesis: verum end; A15: K0 c= the carrier of (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K0 or z in the carrier of (TOP-REAL 2) ) assume z in K0 ; ::_thesis: z in the carrier of (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & p8 `2 <= 0 & p8 <> 0. (TOP-REAL 2) ) ; hence z in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; dom ((cn -FanMorphS) | K0) = (dom (cn -FanMorphS)) /\ K0 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K0 by FUNCT_2:def_1 .= K0 by A15, XBOOLE_1:28 ; then reconsider f = (cn -FanMorphS) | K0 as Function of (((TOP-REAL 2) | D) | K0),((TOP-REAL 2) | D) by A8, A11, FUNCT_2:2, XBOOLE_1:1; A16: K1 = the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; A17: rng ((cn -FanMorphS) | K1) c= the carrier of (((TOP-REAL 2) | D) | K1) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((cn -FanMorphS) | K1) or y in the carrier of (((TOP-REAL 2) | D) | K1) ) assume y in rng ((cn -FanMorphS) | K1) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K1) then consider x being set such that A18: x in dom ((cn -FanMorphS) | K1) and A19: y = ((cn -FanMorphS) | K1) . x by FUNCT_1:def_3; x in (dom (cn -FanMorphS)) /\ K1 by A18, RELAT_1:61; then A20: x in K1 by XBOOLE_0:def_4; K1 c= the carrier of (TOP-REAL 2) by A10, XBOOLE_1:1; then reconsider p = x as Point of (TOP-REAL 2) by A20; (cn -FanMorphS) . p = y by A19, A20, FUNCT_1:49; then y in K1 by A1, A20, Th130; hence y in the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; ::_thesis: verum end; A21: K1 c= the carrier of (TOP-REAL 2) proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in the carrier of (TOP-REAL 2) ) assume z in K1 ; ::_thesis: z in the carrier of (TOP-REAL 2) then ex p8 being Point of (TOP-REAL 2) st ( p8 = z & p8 `2 >= 0 & p8 <> 0. (TOP-REAL 2) ) ; hence z in the carrier of (TOP-REAL 2) ; ::_thesis: verum end; dom ((cn -FanMorphS) | K1) = (dom (cn -FanMorphS)) /\ K1 by RELAT_1:61 .= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1 .= K1 by A21, XBOOLE_1:28 ; then reconsider g = (cn -FanMorphS) | K1 as Function of (((TOP-REAL 2) | D) | K1),((TOP-REAL 2) | D) by A16, A17, FUNCT_2:2, XBOOLE_1:1; A22: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5; A23: D c= K0 \/ K1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in K0 \/ K1 ) assume A24: x in D ; ::_thesis: x in K0 \/ K1 then reconsider px = x as Point of (TOP-REAL 2) ; not x in {(0. (TOP-REAL 2))} by A7, A24, XBOOLE_0:def_5; then ( ( px `2 >= 0 & px <> 0. (TOP-REAL 2) ) or ( px `2 <= 0 & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1; then ( x in K1 or x in K0 ) ; hence x in K0 \/ K1 by XBOOLE_0:def_3; ::_thesis: verum end; A25: dom f = K0 by A8, FUNCT_2:def_1; A26: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5; A27: for x being set st x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) holds f . x = g . x proof let x be set ; ::_thesis: ( x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) implies f . x = g . x ) assume A28: x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) ; ::_thesis: f . x = g . x then x in K0 by A26, XBOOLE_0:def_4; then f . x = (cn -FanMorphS) . x by FUNCT_1:49; hence f . x = g . x by A22, A28, FUNCT_1:49; ::_thesis: verum end; D = [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5; then A29: ([#] (((TOP-REAL 2) | D) | K0)) \/ ([#] (((TOP-REAL 2) | D) | K1)) = [#] ((TOP-REAL 2) | D) by A26, A22, A23, XBOOLE_0:def_10; A30: ( f is continuous & g is continuous ) by A1, A7, Th126, Th127; then consider h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) such that A31: h = f +* g and h is continuous by A26, A22, A29, A9, A27, JGRAPH_2:1; A32: dom g = K1 by A16, FUNCT_2:def_1; ( K0 = [#] (((TOP-REAL 2) | D) | K0) & K1 = [#] (((TOP-REAL 2) | D) | K1) ) by PRE_TOPC:def_5; then A33: f tolerates g by A27, A25, A32, PARTFUN1:def_4; A34: the carrier of ((TOP-REAL 2) | D) = NonZero (TOP-REAL 2) by A7, PRE_TOPC:8; A35: for x being set st x in dom h holds h . x = ((cn -FanMorphS) | D) . x proof let x be set ; ::_thesis: ( x in dom h implies h . x = ((cn -FanMorphS) | D) . x ) assume A36: x in dom h ; ::_thesis: h . x = ((cn -FanMorphS) | D) . x then reconsider p = x as Point of (TOP-REAL 2) by A34, XBOOLE_0:def_5; not x in {(0. (TOP-REAL 2))} by A7, A3, A36, XBOOLE_0:def_5; then A37: x <> 0. (TOP-REAL 2) by TARSKI:def_1; percases ( x in K0 or not x in K0 ) ; supposeA38: x in K0 ; ::_thesis: h . x = ((cn -FanMorphS) | D) . x A39: ((cn -FanMorphS) | D) . p = (cn -FanMorphS) . p by A3, A36, FUNCT_1:49 .= f . p by A38, FUNCT_1:49 ; h . p = (g +* f) . p by A31, A33, FUNCT_4:34 .= f . p by A25, A38, FUNCT_4:13 ; hence h . x = ((cn -FanMorphS) | D) . x by A39; ::_thesis: verum end; suppose not x in K0 ; ::_thesis: h . x = ((cn -FanMorphS) | D) . x then not p `2 <= 0 by A37; then A40: x in K1 by A37; ((cn -FanMorphS) | D) . p = (cn -FanMorphS) . p by A3, A36, FUNCT_1:49 .= g . p by A40, FUNCT_1:49 ; hence h . x = ((cn -FanMorphS) | D) . x by A31, A32, A40, FUNCT_4:13; ::_thesis: verum end; end; end; dom h = the carrier of ((TOP-REAL 2) | D) by FUNCT_2:def_1; then f +* g = (cn -FanMorphS) | D by A31, A4, A35, FUNCT_1:2; hence ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphS) | D & h is continuous ) by A26, A22, A29, A30, A9, A27, JGRAPH_2:1; ::_thesis: verum end; theorem Th132: :: JGRAPH_4:132 for cn being Real st - 1 < cn & cn < 1 holds cn -FanMorphS is continuous proof reconsider D = NonZero (TOP-REAL 2) as non empty Subset of (TOP-REAL 2) by JGRAPH_2:9; let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies cn -FanMorphS is continuous ) assume that A1: - 1 < cn and A2: cn < 1 ; ::_thesis: cn -FanMorphS is continuous reconsider f = cn -FanMorphS as Function of (TOP-REAL 2),(TOP-REAL 2) ; A3: f . (0. (TOP-REAL 2)) = 0. (TOP-REAL 2) by Th113, JGRAPH_2:3; A4: for p being Point of ((TOP-REAL 2) | D) holds f . p <> f . (0. (TOP-REAL 2)) proof let p be Point of ((TOP-REAL 2) | D); ::_thesis: f . p <> f . (0. (TOP-REAL 2)) A5: [#] ((TOP-REAL 2) | D) = D by PRE_TOPC:def_5; then reconsider q = p as Point of (TOP-REAL 2) by XBOOLE_0:def_5; not p in {(0. (TOP-REAL 2))} by A5, XBOOLE_0:def_5; then A6: p <> 0. (TOP-REAL 2) by TARSKI:def_1; percases ( ( (q `1) / |.q.| >= cn & q `2 <= 0 ) or ( (q `1) / |.q.| < cn & q `2 <= 0 ) or q `2 > 0 ) ; supposeA7: ( (q `1) / |.q.| >= cn & q `2 <= 0 ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) set q9 = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]|; A8: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn)) by EUCLID:52; A9: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52; now__::_thesis:_not_|[(|.q.|_*_((((q_`1)_/_|.q.|)_-_cn)_/_(1_-_cn))),(|.q.|_*_(-_(sqrt_(1_-_(((((q_`1)_/_|.q.|)_-_cn)_/_(1_-_cn))_^2)))))]|_=_0._(TOP-REAL_2) assume A10: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction A11: |.q.| <> 0 ^2 by A6, TOPRNS_1:24; then - (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) = - (sqrt (1 - 0)) by A8, A10, JGRAPH_2:3, XCMPLX_1:6 .= - 1 by SQUARE_1:18 ; hence contradiction by A9, A10, A11, JGRAPH_2:3, XCMPLX_1:6; ::_thesis: verum end; hence f . p <> f . (0. (TOP-REAL 2)) by A1, A2, A3, A6, A7, Th115; ::_thesis: verum end; supposeA12: ( (q `1) / |.q.| < cn & q `2 <= 0 ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) set q9 = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]|; A13: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) by EUCLID:52; A14: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))) by EUCLID:52; now__::_thesis:_not_|[(|.q.|_*_((((q_`1)_/_|.q.|)_-_cn)_/_(1_+_cn))),(|.q.|_*_(-_(sqrt_(1_-_(((((q_`1)_/_|.q.|)_-_cn)_/_(1_+_cn))_^2)))))]|_=_0._(TOP-REAL_2) assume A15: |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction A16: |.q.| <> 0 ^2 by A6, TOPRNS_1:24; then - (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) = - (sqrt (1 - 0)) by A13, A15, JGRAPH_2:3, XCMPLX_1:6 .= - 1 by SQUARE_1:18 ; hence contradiction by A14, A15, A16, JGRAPH_2:3, XCMPLX_1:6; ::_thesis: verum end; hence f . p <> f . (0. (TOP-REAL 2)) by A1, A2, A3, A6, A12, Th115; ::_thesis: verum end; suppose q `2 > 0 ; ::_thesis: f . p <> f . (0. (TOP-REAL 2)) then f . p = p by Th113; hence f . p <> f . (0. (TOP-REAL 2)) by A6, Th113, JGRAPH_2:3; ::_thesis: verum end; end; end; A17: for V being Subset of (TOP-REAL 2) st f . (0. (TOP-REAL 2)) in V & V is open holds ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) proof reconsider u0 = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:67; let V be Subset of (TOP-REAL 2); ::_thesis: ( f . (0. (TOP-REAL 2)) in V & V is open implies ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) ) reconsider VV = V as Subset of (TopSpaceMetr (Euclid 2)) by Lm11; assume that A18: f . (0. (TOP-REAL 2)) in V and A19: V is open ; ::_thesis: ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) VV is open by A19, Lm11, PRE_TOPC:30; then consider r being real number such that A20: r > 0 and A21: Ball (u0,r) c= V by A3, A18, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider W1 = Ball (u0,r) as Subset of (TOP-REAL 2) ; A22: W1 is open by GOBOARD6:3; A23: f .: W1 c= W1 proof let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f .: W1 or z in W1 ) assume z in f .: W1 ; ::_thesis: z in W1 then consider y being set such that A24: y in dom f and A25: y in W1 and A26: z = f . y by FUNCT_1:def_6; z in rng f by A24, A26, FUNCT_1:def_3; then reconsider qz = z as Point of (TOP-REAL 2) ; reconsider pz = qz as Point of (Euclid 2) by EUCLID:67; reconsider q = y as Point of (TOP-REAL 2) by A24; reconsider qy = q as Point of (Euclid 2) by EUCLID:67; dist (u0,qy) < r by A25, METRIC_1:11; then A27: |.((0. (TOP-REAL 2)) - q).| < r by JGRAPH_1:28; percases ( q `2 >= 0 or ( q <> 0. (TOP-REAL 2) & (q `1) / |.q.| >= cn & q `2 <= 0 ) or ( q <> 0. (TOP-REAL 2) & (q `1) / |.q.| < cn & q `2 <= 0 ) ) by JGRAPH_2:3; suppose q `2 >= 0 ; ::_thesis: z in W1 hence z in W1 by A25, A26, Th113; ::_thesis: verum end; supposeA28: ( q <> 0. (TOP-REAL 2) & (q `1) / |.q.| >= cn & q `2 <= 0 ) ; ::_thesis: z in W1 then A29: ((q `1) / |.q.|) - cn >= 0 by XREAL_1:48; 0 <= (q `2) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A30: ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; A31: 1 - cn > 0 by A2, XREAL_1:149; |.q.| <> 0 by A28, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A30, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (q `1) / |.q.| by SQUARE_1:51; then 1 - cn >= ((q `1) / |.q.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((q `1) / |.q.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A31, XREAL_1:72; then - 1 <= (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A31, XCMPLX_1:197; then ((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A31, A29, SQUARE_1:49; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A32: 1 - ((- ((((q `1) / |.q.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; A33: (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by A1, A2, A28, Th115; then A34: qz `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn)) by A26, EUCLID:52; qz `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))) by A26, A33, EUCLID:52; then A35: (qz `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)) by A32, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A34, A35 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; then A36: |.qz.| = |.q.| by SQUARE_1:22; |.(- q).| < r by A27, EUCLID:27; then |.q.| < r by TOPRNS_1:26; then |.(- qz).| < r by A36, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - qz).| < r by EUCLID:27; then dist (u0,pz) < r by JGRAPH_1:28; hence z in W1 by METRIC_1:11; ::_thesis: verum end; supposeA37: ( q <> 0. (TOP-REAL 2) & (q `1) / |.q.| < cn & q `2 <= 0 ) ; ::_thesis: z in W1 0 <= (q `2) ^2 by XREAL_1:63; then ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) <= ((q `1) ^2) + ((q `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A38: ((q `1) ^2) / (|.q.| ^2) <= (|.q.| ^2) / (|.q.| ^2) by XREAL_1:72; A39: 1 + cn > 0 by A1, XREAL_1:148; |.q.| <> 0 by A37, TOPRNS_1:24; then |.q.| ^2 > 0 by SQUARE_1:12; then ((q `1) ^2) / (|.q.| ^2) <= 1 by A38, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (q `1) / |.q.| by SQUARE_1:51; then - (- 1) >= - ((q `1) / |.q.|) by XREAL_1:24; then 1 + cn >= (- ((q `1) / |.q.|)) + cn by XREAL_1:7; then A40: (- (((q `1) / |.q.|) - cn)) / (1 + cn) <= 1 by A39, XREAL_1:185; cn - ((q `1) / |.q.|) >= 0 by A37, XREAL_1:48; then - 1 <= (- (((q `1) / |.q.|) - cn)) / (1 + cn) by A39; then ((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A40, SQUARE_1:49; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then A41: 1 - ((- ((((q `1) / |.q.|) - cn) / (1 + cn))) ^2) >= 0 by XCMPLX_1:187; A42: (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| by A1, A2, A37, Th115; then A43: qz `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) by A26, EUCLID:52; qz `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))) by A26, A42, EUCLID:52; then A44: (qz `2) ^2 = (|.q.| ^2) * ((sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) ^2) .= (|.q.| ^2) * (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)) by A41, SQUARE_1:def_2 ; |.qz.| ^2 = ((qz `1) ^2) + ((qz `2) ^2) by JGRAPH_3:1 .= |.q.| ^2 by A43, A44 ; then sqrt (|.qz.| ^2) = |.q.| by SQUARE_1:22; then A45: |.qz.| = |.q.| by SQUARE_1:22; |.(- q).| < r by A27, EUCLID:27; then |.q.| < r by TOPRNS_1:26; then |.(- qz).| < r by A45, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - qz).| < r by EUCLID:27; then dist (u0,pz) < r by JGRAPH_1:28; hence z in W1 by METRIC_1:11; ::_thesis: verum end; end; end; u0 in W1 by A20, GOBOARD6:1; hence ex W being Subset of (TOP-REAL 2) st ( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) by A21, A22, A23, XBOOLE_1:1; ::_thesis: verum end; A46: D ` = {(0. (TOP-REAL 2))} by JGRAPH_3:20; then ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (cn -FanMorphS) | D & h is continuous ) by A1, A2, Th131; hence cn -FanMorphS is continuous by A3, A46, A4, A17, JGRAPH_3:3; ::_thesis: verum end; theorem Th133: :: JGRAPH_4:133 for cn being Real st - 1 < cn & cn < 1 holds cn -FanMorphS is one-to-one proof let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies cn -FanMorphS is one-to-one ) assume that A1: - 1 < cn and A2: cn < 1 ; ::_thesis: cn -FanMorphS is one-to-one for x1, x2 being set st x1 in dom (cn -FanMorphS) & x2 in dom (cn -FanMorphS) & (cn -FanMorphS) . x1 = (cn -FanMorphS) . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in dom (cn -FanMorphS) & x2 in dom (cn -FanMorphS) & (cn -FanMorphS) . x1 = (cn -FanMorphS) . x2 implies x1 = x2 ) assume that A3: x1 in dom (cn -FanMorphS) and A4: x2 in dom (cn -FanMorphS) and A5: (cn -FanMorphS) . x1 = (cn -FanMorphS) . x2 ; ::_thesis: x1 = x2 reconsider p2 = x2 as Point of (TOP-REAL 2) by A4; reconsider p1 = x1 as Point of (TOP-REAL 2) by A3; set q = p1; set p = p2; A6: 1 - cn > 0 by A2, XREAL_1:149; percases ( p1 `2 >= 0 or ( (p1 `1) / |.p1.| >= cn & p1 `2 <= 0 & p1 <> 0. (TOP-REAL 2) ) or ( (p1 `1) / |.p1.| < cn & p1 `2 <= 0 & p1 <> 0. (TOP-REAL 2) ) ) by JGRAPH_2:3; supposeA7: p1 `2 >= 0 ; ::_thesis: x1 = x2 then A8: (cn -FanMorphS) . p1 = p1 by Th113; percases ( p2 `2 >= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 <= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 <= 0 ) ) by JGRAPH_2:3; suppose p2 `2 >= 0 ; ::_thesis: x1 = x2 hence x1 = x2 by A5, A8, Th113; ::_thesis: verum end; supposeA9: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 <= 0 ) ; ::_thesis: x1 = x2 then A10: |.p2.| <> 0 by TOPRNS_1:24; then A11: |.p2.| ^2 > 0 by SQUARE_1:12; A12: ((p2 `1) / |.p2.|) - cn >= 0 by A9, XREAL_1:48; A13: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; 0 <= (p2 `2) ^2 by XREAL_1:63; then 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) by XREAL_1:7; then ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by A13, XREAL_1:72; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A11, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p2 `1) / |.p2.| by SQUARE_1:51; then 1 - cn >= ((p2 `1) / |.p2.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((p2 `1) / |.p2.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p2 `1) / |.p2.|) - cn)) / (1 - cn) by A6, XREAL_1:72; then A14: - 1 <= (- (((p2 `1) / |.p2.|) - cn)) / (1 - cn) by A6, XCMPLX_1:197; A15: (cn -FanMorphS) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| by A1, A2, A9, Th115; then A16: p1 `2 = |.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))) by A5, A8, EUCLID:52; ((p2 `1) / |.p2.|) - cn >= 0 by A9, XREAL_1:48; then ((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A6, A14, SQUARE_1:49; then A17: 1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p2 `1) / |.p2.|) - cn)) ^2) / ((1 - cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p2 `1) / |.p2.|) - cn) ^2) / ((1 - cn) ^2))) >= 0 ; then sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)) >= 0 by XCMPLX_1:76; then p1 `2 = 0 by A5, A7, A8, A15, EUCLID:52; then A18: - (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))) = - 0 by A16, A10, XCMPLX_1:6; 1 - ((- ((((p2 `1) / |.p2.|) - cn) / (1 - cn))) ^2) >= 0 by A17, XCMPLX_1:187; then 1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2) = 0 by A18, SQUARE_1:24; then 1 = (((p2 `1) / |.p2.|) - cn) / (1 - cn) by A6, A12, SQUARE_1:18, SQUARE_1:22; then 1 * (1 - cn) = ((p2 `1) / |.p2.|) - cn by A6, XCMPLX_1:87; then 1 * |.p2.| = p2 `1 by A9, TOPRNS_1:24, XCMPLX_1:87; then p2 `2 = 0 by A13, XCMPLX_1:6; hence x1 = x2 by A5, A8, Th113; ::_thesis: verum end; supposeA19: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 <= 0 ) ; ::_thesis: x1 = x2 then A20: (cn -FanMorphS) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| by A1, A2, Th115; then A21: p1 `2 = |.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))) by A5, A8, EUCLID:52; A22: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1; A23: |.p2.| <> 0 by A19, TOPRNS_1:24; then A24: |.p2.| ^2 > 0 by SQUARE_1:12; A25: 1 + cn > 0 by A1, XREAL_1:148; A26: ((p2 `1) / |.p2.|) - cn <= 0 by A19, XREAL_1:47; then A27: - 1 <= (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) by A25; 0 <= (p2 `2) ^2 by XREAL_1:63; then 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) by XREAL_1:7; then ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by A22, XREAL_1:72; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A24, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (- ((p2 `1) / |.p2.|)) ^2 <= 1 ; then 1 >= - ((p2 `1) / |.p2.|) by SQUARE_1:51; then 1 + cn >= (- ((p2 `1) / |.p2.|)) + cn by XREAL_1:7; then (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) <= 1 by A25, XREAL_1:185; then ((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A27, SQUARE_1:49; then A28: 1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p2 `1) / |.p2.|) - cn)) ^2) / ((1 + cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p2 `1) / |.p2.|) - cn) ^2) / ((1 + cn) ^2))) >= 0 ; then sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)) >= 0 by XCMPLX_1:76; then p1 `2 = 0 by A5, A7, A8, A20, EUCLID:52; then A29: - (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))) = - 0 by A21, A23, XCMPLX_1:6; 1 - ((- ((((p2 `1) / |.p2.|) - cn) / (1 + cn))) ^2) >= 0 by A28, XCMPLX_1:187; then 1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2) = 0 by A29, SQUARE_1:24; then 1 = (- ((((p2 `1) / |.p2.|) - cn) / (1 + cn))) ^2 ; then 1 = - ((((p2 `1) / |.p2.|) - cn) / (1 + cn)) by A25, A26, SQUARE_1:18, SQUARE_1:22; then 1 = (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) by XCMPLX_1:187; then 1 * (1 + cn) = - (((p2 `1) / |.p2.|) - cn) by A25, XCMPLX_1:87; then (1 + cn) - cn = - ((p2 `1) / |.p2.|) ; then 1 = (- (p2 `1)) / |.p2.| by XCMPLX_1:187; then 1 * |.p2.| = - (p2 `1) by A19, TOPRNS_1:24, XCMPLX_1:87; then ((p2 `1) ^2) - ((p2 `1) ^2) = (p2 `2) ^2 by A22, XCMPLX_1:26; then p2 `2 = 0 by XCMPLX_1:6; hence x1 = x2 by A5, A8, Th113; ::_thesis: verum end; end; end; supposeA30: ( (p1 `1) / |.p1.| >= cn & p1 `2 <= 0 & p1 <> 0. (TOP-REAL 2) ) ; ::_thesis: x1 = x2 then |.p1.| <> 0 by TOPRNS_1:24; then A31: |.p1.| ^2 > 0 by SQUARE_1:12; set q4 = |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]|; A32: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]| `1 = |.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn)) by EUCLID:52; A33: (cn -FanMorphS) . p1 = |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]| by A1, A2, A30, Th115; percases ( p2 `2 >= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 <= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 <= 0 ) ) by JGRAPH_2:3; supposeA34: p2 `2 >= 0 ; ::_thesis: x1 = x2 then A35: (cn -FanMorphS) . p2 = p2 by Th113; then A36: p2 `2 = |.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))) by A5, A33, EUCLID:52; A37: ((p1 `1) / |.p1.|) - cn >= 0 by A30, XREAL_1:48; A38: 1 - cn > 0 by A2, XREAL_1:149; A39: |.p1.| <> 0 by A30, TOPRNS_1:24; then A40: |.p1.| ^2 > 0 by SQUARE_1:12; A41: ((p1 `1) / |.p1.|) - cn >= 0 by A30, XREAL_1:48; A42: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1; 0 <= (p1 `2) ^2 by XREAL_1:63; then 0 + ((p1 `1) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) by XREAL_1:7; then ((p1 `1) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by A42, XREAL_1:72; then ((p1 `1) ^2) / (|.p1.| ^2) <= 1 by A40, XCMPLX_1:60; then ((p1 `1) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p1 `1) / |.p1.| by SQUARE_1:51; then 1 - cn >= ((p1 `1) / |.p1.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((p1 `1) / |.p1.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p1 `1) / |.p1.|) - cn)) / (1 - cn) by A38, XREAL_1:72; then - 1 <= (- (((p1 `1) / |.p1.|) - cn)) / (1 - cn) by A38, XCMPLX_1:197; then ((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A38, A41, SQUARE_1:49; then A43: 1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then sqrt (1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2)) >= 0 by SQUARE_1:def_2; then sqrt (1 - (((- (((p1 `1) / |.p1.|) - cn)) ^2) / ((1 - cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) ^2) / ((1 - cn) ^2))) >= 0 ; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)) >= 0 by XCMPLX_1:76; then p2 `2 = 0 by A5, A33, A34, A35, EUCLID:52; then A44: - (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))) = - 0 by A36, A39, XCMPLX_1:6; 1 - ((- ((((p1 `1) / |.p1.|) - cn) / (1 - cn))) ^2) >= 0 by A43, XCMPLX_1:187; then 1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2) = 0 by A44, SQUARE_1:24; then 1 = (((p1 `1) / |.p1.|) - cn) / (1 - cn) by A38, A37, SQUARE_1:18, SQUARE_1:22; then 1 * (1 - cn) = ((p1 `1) / |.p1.|) - cn by A38, XCMPLX_1:87; then 1 * |.p1.| = p1 `1 by A30, TOPRNS_1:24, XCMPLX_1:87; then p1 `2 = 0 by A42, XCMPLX_1:6; hence x1 = x2 by A5, A35, Th113; ::_thesis: verum end; supposeA45: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 <= 0 ) ; ::_thesis: x1 = x2 0 <= (p1 `2) ^2 by XREAL_1:63; then ( |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) & 0 + ((p1 `1) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p1 `1) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by XREAL_1:72; then ((p1 `1) ^2) / (|.p1.| ^2) <= 1 by A31, XCMPLX_1:60; then ((p1 `1) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p1 `1) / |.p1.| by SQUARE_1:51; then 1 - cn >= ((p1 `1) / |.p1.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((p1 `1) / |.p1.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p1 `1) / |.p1.|) - cn)) / (1 - cn) by A6, XREAL_1:72; then A46: - 1 <= (- (((p1 `1) / |.p1.|) - cn)) / (1 - cn) by A6, XCMPLX_1:197; ((p1 `1) / |.p1.|) - cn >= 0 by A30, XREAL_1:48; then ((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A6, A46, SQUARE_1:49; then 1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A47: 1 - ((- ((((p1 `1) / |.p1.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]| `2 = |.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52; then A48: (|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]| `2) ^2 = (|.p1.| ^2) * ((sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2))) ^2) .= (|.p1.| ^2) * (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)) by A47, SQUARE_1:def_2 ; A49: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]| `1 = |.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn)) by EUCLID:52; |.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]|.| ^2 = ((|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]| `1) ^2) + ((|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]| `2) ^2) by JGRAPH_3:1 .= |.p1.| ^2 by A49, A48 ; then A50: sqrt (|.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]|.| ^2) = |.p1.| by SQUARE_1:22; then A51: |.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 - cn)) ^2)))))]|.| = |.p1.| by SQUARE_1:22; 0 <= (p2 `2) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A52: ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; |.p2.| <> 0 by A45, TOPRNS_1:24; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A52, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then 1 >= (p2 `1) / |.p2.| by SQUARE_1:51; then 1 - cn >= ((p2 `1) / |.p2.|) - cn by XREAL_1:9; then - (1 - cn) <= - (((p2 `1) / |.p2.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) <= (- (((p2 `1) / |.p2.|) - cn)) / (1 - cn) by A6, XREAL_1:72; then A53: - 1 <= (- (((p2 `1) / |.p2.|) - cn)) / (1 - cn) by A6, XCMPLX_1:197; ((p2 `1) / |.p2.|) - cn >= 0 by A45, XREAL_1:48; then ((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2 <= 1 ^2 by A6, A53, SQUARE_1:49; then 1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 - cn)) ^2) >= 0 by XREAL_1:48; then A54: 1 - ((- ((((p2 `1) / |.p2.|) - cn) / (1 - cn))) ^2) >= 0 by XCMPLX_1:187; set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]|; A55: |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn)) by EUCLID:52; |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| `2 = |.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52; then A56: (|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| `2) ^2 = (|.p2.| ^2) * ((sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2))) ^2) .= (|.p2.| ^2) * (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)) by A54, SQUARE_1:def_2 ; |.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]|.| ^2 = ((|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| `1) ^2) + ((|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| `2) ^2) by JGRAPH_3:1 .= |.p2.| ^2 by A55, A56 ; then A57: sqrt (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]|.| ^2) = |.p2.| by SQUARE_1:22; then A58: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]|.| = |.p2.| by SQUARE_1:22; A59: (cn -FanMorphS) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| by A1, A2, A45, Th115; then (((p2 `1) / |.p2.|) - cn) / (1 - cn) = (|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 - cn))) / |.p2.| by A5, A33, A32, A45, A55, TOPRNS_1:24, XCMPLX_1:89; then (((p2 `1) / |.p2.|) - cn) / (1 - cn) = (((p1 `1) / |.p1.|) - cn) / (1 - cn) by A5, A33, A45, A59, A50, A57, TOPRNS_1:24, XCMPLX_1:89; then ((((p2 `1) / |.p2.|) - cn) / (1 - cn)) * (1 - cn) = ((p1 `1) / |.p1.|) - cn by A6, XCMPLX_1:87; then ((p2 `1) / |.p2.|) - cn = ((p1 `1) / |.p1.|) - cn by A6, XCMPLX_1:87; then ((p2 `1) / |.p2.|) * |.p2.| = p1 `1 by A5, A33, A45, A59, A51, A58, TOPRNS_1:24, XCMPLX_1:87; then A60: p2 `1 = p1 `1 by A45, TOPRNS_1:24, XCMPLX_1:87; ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1; then (- (p2 `2)) ^2 = (p1 `2) ^2 by A5, A33, A59, A51, A58, A60; then - (p2 `2) = sqrt ((- (p1 `2)) ^2) by A45, SQUARE_1:22; then A61: - (- (p2 `2)) = - (- (p1 `2)) by A30, SQUARE_1:22; p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53; hence x1 = x2 by A60, A61, EUCLID:53; ::_thesis: verum end; supposeA62: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 <= 0 ) ; ::_thesis: x1 = x2 then ((p2 `1) / |.p2.|) - cn < 0 by XREAL_1:49; then A63: (((p2 `1) / |.p2.|) - cn) / (1 + cn) < 0 by A1, XREAL_1:141, XREAL_1:148; set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]|; A64: ( |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn)) & ((p1 `1) / |.p1.|) - cn >= 0 ) by A30, EUCLID:52, XREAL_1:48; A65: 1 - cn > 0 by A2, XREAL_1:149; ( (cn -FanMorphS) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| & |.p2.| <> 0 ) by A1, A2, A62, Th115, TOPRNS_1:24; hence x1 = x2 by A5, A33, A32, A63, A64, A65, XREAL_1:132; ::_thesis: verum end; end; end; supposeA66: ( (p1 `1) / |.p1.| < cn & p1 `2 <= 0 & p1 <> 0. (TOP-REAL 2) ) ; ::_thesis: x1 = x2 then A67: |.p1.| <> 0 by TOPRNS_1:24; then A68: |.p1.| ^2 > 0 by SQUARE_1:12; set q4 = |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]|; A69: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]| `1 = |.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn)) by EUCLID:52; A70: (cn -FanMorphS) . p1 = |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]| by A1, A2, A66, Th115; percases ( p2 `2 >= 0 or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 <= 0 ) or ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 <= 0 ) ) by JGRAPH_2:3; supposeA71: p2 `2 >= 0 ; ::_thesis: x1 = x2 then A72: (cn -FanMorphS) . p2 = p2 by Th113; then A73: p2 `2 = |.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))) by A5, A70, EUCLID:52; A74: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1; A75: 1 + cn > 0 by A1, XREAL_1:148; 0 <= (p1 `2) ^2 by XREAL_1:63; then 0 + ((p1 `1) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) by XREAL_1:7; then ((p1 `1) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by A74, XREAL_1:72; then ((p1 `1) ^2) / (|.p1.| ^2) <= 1 by A68, XCMPLX_1:60; then ((p1 `1) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then (- ((p1 `1) / |.p1.|)) ^2 <= 1 ; then 1 >= - ((p1 `1) / |.p1.|) by SQUARE_1:51; then 1 + cn >= (- ((p1 `1) / |.p1.|)) + cn by XREAL_1:7; then A76: (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) <= 1 by A75, XREAL_1:185; A77: ((p1 `1) / |.p1.|) - cn <= 0 by A66, XREAL_1:47; then - 1 <= (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) by A75; then ((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A76, SQUARE_1:49; then A78: 1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then A79: 1 - ((- ((((p1 `1) / |.p1.|) - cn) / (1 + cn))) ^2) >= 0 by XCMPLX_1:187; sqrt (1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2)) >= 0 by A78, SQUARE_1:def_2; then sqrt (1 - (((- (((p1 `1) / |.p1.|) - cn)) ^2) / ((1 + cn) ^2))) >= 0 by XCMPLX_1:76; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) ^2) / ((1 + cn) ^2))) >= 0 ; then sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)) >= 0 by XCMPLX_1:76; then p2 `2 = 0 by A5, A70, A71, A72, EUCLID:52; then - (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))) = - 0 by A67, A73, XCMPLX_1:6; then 1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2) = 0 by A79, SQUARE_1:24; then 1 = (- ((((p1 `1) / |.p1.|) - cn) / (1 + cn))) ^2 ; then 1 = - ((((p1 `1) / |.p1.|) - cn) / (1 + cn)) by A75, A77, SQUARE_1:18, SQUARE_1:22; then 1 = (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) by XCMPLX_1:187; then 1 * (1 + cn) = - (((p1 `1) / |.p1.|) - cn) by A75, XCMPLX_1:87; then (1 + cn) - cn = - ((p1 `1) / |.p1.|) ; then 1 = (- (p1 `1)) / |.p1.| by XCMPLX_1:187; then 1 * |.p1.| = - (p1 `1) by A66, TOPRNS_1:24, XCMPLX_1:87; then ((p1 `1) ^2) - ((p1 `1) ^2) = (p1 `2) ^2 by A74, XCMPLX_1:26; then p1 `2 = 0 by XCMPLX_1:6; hence x1 = x2 by A5, A72, Th113; ::_thesis: verum end; supposeA80: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| >= cn & p2 `2 <= 0 ) ; ::_thesis: x1 = x2 set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]|; A81: ( |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn)) & |.p1.| <> 0 ) by A66, EUCLID:52, TOPRNS_1:24; ((p1 `1) / |.p1.|) - cn < 0 by A66, XREAL_1:49; then A82: (((p1 `1) / |.p1.|) - cn) / (1 + cn) < 0 by A1, XREAL_1:141, XREAL_1:148; A83: 1 - cn > 0 by A2, XREAL_1:149; ( (cn -FanMorphS) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 - cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 - cn)) ^2)))))]| & ((p2 `1) / |.p2.|) - cn >= 0 ) by A1, A2, A80, Th115, XREAL_1:48; hence x1 = x2 by A5, A70, A69, A82, A81, A83, XREAL_1:132; ::_thesis: verum end; supposeA84: ( p2 <> 0. (TOP-REAL 2) & (p2 `1) / |.p2.| < cn & p2 `2 <= 0 ) ; ::_thesis: x1 = x2 0 <= (p2 `2) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then A85: ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; A86: 1 + cn > 0 by A1, XREAL_1:148; 0 <= (p1 `2) ^2 by XREAL_1:63; then ( |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) & 0 + ((p1 `1) ^2) <= ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p1 `1) ^2) / (|.p1.| ^2) <= (|.p1.| ^2) / (|.p1.| ^2) by XREAL_1:72; then ((p1 `1) ^2) / (|.p1.| ^2) <= 1 by A68, XCMPLX_1:60; then ((p1 `1) / |.p1.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p1 `1) / |.p1.| by SQUARE_1:51; then (- 1) - cn <= ((p1 `1) / |.p1.|) - cn by XREAL_1:9; then - ((- 1) - cn) >= - (((p1 `1) / |.p1.|) - cn) by XREAL_1:24; then A87: (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) <= 1 by A86, XREAL_1:185; ((p1 `1) / |.p1.|) - cn <= 0 by A66, XREAL_1:47; then - 1 <= (- (((p1 `1) / |.p1.|) - cn)) / (1 + cn) by A86; then ((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A87, SQUARE_1:49; then 1 - (((- (((p1 `1) / |.p1.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then A88: 1 - ((- ((((p1 `1) / |.p1.|) - cn) / (1 + cn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]| `2 = |.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))) by EUCLID:52; then A89: (|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]| `2) ^2 = (|.p1.| ^2) * ((sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2))) ^2) .= (|.p1.| ^2) * (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)) by A88, SQUARE_1:def_2 ; A90: |[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]| `1 = |.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn)) by EUCLID:52; set p4 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]|; A91: |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn)) by EUCLID:52; |.p2.| <> 0 by A84, TOPRNS_1:24; then |.p2.| ^2 > 0 by SQUARE_1:12; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A85, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then - 1 <= (p2 `1) / |.p2.| by SQUARE_1:51; then (- 1) - cn <= ((p2 `1) / |.p2.|) - cn by XREAL_1:9; then - ((- 1) - cn) >= - (((p2 `1) / |.p2.|) - cn) by XREAL_1:24; then A92: (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) <= 1 by A86, XREAL_1:185; ((p2 `1) / |.p2.|) - cn <= 0 by A84, XREAL_1:47; then - 1 <= (- (((p2 `1) / |.p2.|) - cn)) / (1 + cn) by A86; then ((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2 <= 1 ^2 by A92, SQUARE_1:49; then 1 - (((- (((p2 `1) / |.p2.|) - cn)) / (1 + cn)) ^2) >= 0 by XREAL_1:48; then A93: 1 - ((- ((((p2 `1) / |.p2.|) - cn) / (1 + cn))) ^2) >= 0 by XCMPLX_1:187; |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| `2 = |.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))) by EUCLID:52; then A94: (|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| `2) ^2 = (|.p2.| ^2) * ((sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2))) ^2) .= (|.p2.| ^2) * (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)) by A93, SQUARE_1:def_2 ; |.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]|.| ^2 = ((|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| `1) ^2) + ((|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| `2) ^2) by JGRAPH_3:1 .= |.p2.| ^2 by A91, A94 ; then A95: sqrt (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]|.| ^2) = |.p2.| by SQUARE_1:22; then A96: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]|.| = |.p2.| by SQUARE_1:22; |.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]|.| ^2 = ((|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]| `1) ^2) + ((|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]| `2) ^2) by JGRAPH_3:1 .= |.p1.| ^2 by A90, A89 ; then A97: sqrt (|.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]|.| ^2) = |.p1.| by SQUARE_1:22; then A98: |.|[(|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))),(|.p1.| * (- (sqrt (1 - (((((p1 `1) / |.p1.|) - cn) / (1 + cn)) ^2)))))]|.| = |.p1.| by SQUARE_1:22; A99: (cn -FanMorphS) . p2 = |[(|.p2.| * ((((p2 `1) / |.p2.|) - cn) / (1 + cn))),(|.p2.| * (- (sqrt (1 - (((((p2 `1) / |.p2.|) - cn) / (1 + cn)) ^2)))))]| by A1, A2, A84, Th115; then (((p2 `1) / |.p2.|) - cn) / (1 + cn) = (|.p1.| * ((((p1 `1) / |.p1.|) - cn) / (1 + cn))) / |.p2.| by A5, A70, A69, A84, A91, TOPRNS_1:24, XCMPLX_1:89; then (((p2 `1) / |.p2.|) - cn) / (1 + cn) = (((p1 `1) / |.p1.|) - cn) / (1 + cn) by A5, A70, A84, A99, A97, A95, TOPRNS_1:24, XCMPLX_1:89; then ((((p2 `1) / |.p2.|) - cn) / (1 + cn)) * (1 + cn) = ((p1 `1) / |.p1.|) - cn by A86, XCMPLX_1:87; then ((p2 `1) / |.p2.|) - cn = ((p1 `1) / |.p1.|) - cn by A86, XCMPLX_1:87; then ((p2 `1) / |.p2.|) * |.p2.| = p1 `1 by A5, A70, A84, A99, A98, A96, TOPRNS_1:24, XCMPLX_1:87; then A100: p2 `1 = p1 `1 by A84, TOPRNS_1:24, XCMPLX_1:87; ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) ) by JGRAPH_3:1; then (- (p2 `2)) ^2 = (p1 `2) ^2 by A5, A70, A99, A98, A96, A100; then - (p2 `2) = sqrt ((- (p1 `2)) ^2) by A84, SQUARE_1:22; then A101: - (- (p2 `2)) = - (- (p1 `2)) by A66, SQUARE_1:22; p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53; hence x1 = x2 by A100, A101, EUCLID:53; ::_thesis: verum end; end; end; end; end; hence cn -FanMorphS is one-to-one by FUNCT_1:def_4; ::_thesis: verum end; theorem Th134: :: JGRAPH_4:134 for cn being Real st - 1 < cn & cn < 1 holds ( cn -FanMorphS is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) ) proof let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies ( cn -FanMorphS is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) ) ) assume that A1: - 1 < cn and A2: cn < 1 ; ::_thesis: ( cn -FanMorphS is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) ) thus cn -FanMorphS is Function of (TOP-REAL 2),(TOP-REAL 2) ; ::_thesis: rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = cn -FanMorphS holds rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) proof let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = cn -FanMorphS implies rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) ) assume A3: f = cn -FanMorphS ; ::_thesis: rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) A4: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; the carrier of (TOP-REAL 2) c= rng f proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (TOP-REAL 2) or y in rng f ) assume y in the carrier of (TOP-REAL 2) ; ::_thesis: y in rng f then reconsider p2 = y as Point of (TOP-REAL 2) ; set q = p2; now__::_thesis:_(_(_p2_`2_>=_0_&_ex_x_being_set_st_ (_x_in_dom_(cn_-FanMorphS)_&_y_=_(cn_-FanMorphS)_._x_)_)_or_(_(p2_`1)_/_|.p2.|_>=_0_&_p2_`2_<=_0_&_p2_<>_0._(TOP-REAL_2)_&_ex_x_being_set_st_ (_x_in_dom_(cn_-FanMorphS)_&_y_=_(cn_-FanMorphS)_._x_)_)_or_(_(p2_`1)_/_|.p2.|_<_0_&_p2_`2_<=_0_&_p2_<>_0._(TOP-REAL_2)_&_ex_x_being_set_st_ (_x_in_dom_(cn_-FanMorphS)_&_y_=_(cn_-FanMorphS)_._x_)_)_) percases ( p2 `2 >= 0 or ( (p2 `1) / |.p2.| >= 0 & p2 `2 <= 0 & p2 <> 0. (TOP-REAL 2) ) or ( (p2 `1) / |.p2.| < 0 & p2 `2 <= 0 & p2 <> 0. (TOP-REAL 2) ) ) by JGRAPH_2:3; case p2 `2 >= 0 ; ::_thesis: ex x being set st ( x in dom (cn -FanMorphS) & y = (cn -FanMorphS) . x ) then y = (cn -FanMorphS) . p2 by Th113; hence ex x being set st ( x in dom (cn -FanMorphS) & y = (cn -FanMorphS) . x ) by A3, A4; ::_thesis: verum end; caseA5: ( (p2 `1) / |.p2.| >= 0 & p2 `2 <= 0 & p2 <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st ( x in dom (cn -FanMorphS) & y = (cn -FanMorphS) . x ) - (- (1 + cn)) > 0 by A1, XREAL_1:148; then A6: - ((- 1) - cn) > 0 ; A7: 1 - cn >= 0 by A2, XREAL_1:149; then ((p2 `1) / |.p2.|) * (1 - cn) >= 0 by A5; then (- 1) - cn <= ((p2 `1) / |.p2.|) * (1 - cn) by A6; then A8: ((- 1) - cn) + cn <= (((p2 `1) / |.p2.|) * (1 - cn)) + cn by XREAL_1:7; set px = |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|; A9: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn) by EUCLID:52; |.p2.| <> 0 by A5, TOPRNS_1:24; then A10: |.p2.| ^2 > 0 by SQUARE_1:12; A11: dom (cn -FanMorphS) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; A12: 1 - cn > 0 by A2, XREAL_1:149; 0 <= (p2 `2) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A10, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (p2 `1) / |.p2.| <= 1 by SQUARE_1:51; then ((p2 `1) / |.p2.|) * (1 - cn) <= 1 * (1 - cn) by A12, XREAL_1:64; then ((((p2 `1) / |.p2.|) * (1 - cn)) + cn) - cn <= 1 - cn ; then (((p2 `1) / |.p2.|) * (1 - cn)) + cn <= 1 by XREAL_1:9; then 1 ^2 >= ((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2 by A8, SQUARE_1:49; then A13: 1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2) >= 0 by XREAL_1:48; then A14: sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)) >= 0 by SQUARE_1:def_2; A15: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| `2 = - (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))) by EUCLID:52; then |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2 = (((- |.p2.|) * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))) ^2) + ((|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)) ^2) by A9, JGRAPH_3:1 .= ((|.p2.| ^2) * ((sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))) ^2)) + ((|.p2.| ^2) * (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)) ; then A16: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2 = ((|.p2.| ^2) * (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2))) + ((|.p2.| ^2) * (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)) by A13, SQUARE_1:def_2 .= |.p2.| ^2 ; then A17: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| = sqrt (|.p2.| ^2) by SQUARE_1:22 .= |.p2.| by SQUARE_1:22 ; then A18: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| <> 0. (TOP-REAL 2) by A5, TOPRNS_1:23, TOPRNS_1:24; (((p2 `1) / |.p2.|) * (1 - cn)) + cn >= 0 + cn by A5, A7, XREAL_1:7; then (|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| >= cn by A5, A9, A17, TOPRNS_1:24, XCMPLX_1:89; then A19: (cn -FanMorphS) . |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| = |[(|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * ((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.|) - cn) / (1 - cn))),(|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt (1 - (((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.|) - cn) / (1 - cn)) ^2)))))]| by A1, A2, A15, A14, A18, Th115; |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (sqrt ((- ((p2 `2) / |.p2.|)) ^2)) = |.p2.| * (- ((p2 `2) / |.p2.|)) by A5, A17, SQUARE_1:22 .= ((- (p2 `2)) / |.p2.|) * |.p2.| by XCMPLX_1:187 .= - (p2 `2) by A5, TOPRNS_1:24, XCMPLX_1:87 ; then A20: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt ((- ((p2 `2) / |.p2.|)) ^2))) = p2 `2 ; A21: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * ((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.|) - cn) / (1 - cn)) = |.p2.| * ((((((p2 `1) / |.p2.|) * (1 - cn)) + cn) - cn) / (1 - cn)) by A5, A9, A17, TOPRNS_1:24, XCMPLX_1:89 .= |.p2.| * ((p2 `1) / |.p2.|) by A12, XCMPLX_1:89 .= p2 `1 by A5, TOPRNS_1:24, XCMPLX_1:87 ; then |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt (1 - (((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.|) - cn) / (1 - cn)) ^2)))) = |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt (1 - (((p2 `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.|) ^2)))) by A5, A17, TOPRNS_1:24, XCMPLX_1:89 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt (1 - (((p2 `1) ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2))))) by XCMPLX_1:76 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt (((|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2)) - (((p2 `1) ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2))))) by A10, A16, XCMPLX_1:60 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt (((|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2) - ((p2 `1) ^2)) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2)))) by XCMPLX_1:120 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt (((((p2 `1) ^2) + ((p2 `2) ^2)) - ((p2 `1) ^2)) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| ^2)))) by A16, JGRAPH_3:1 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 - cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 - cn)) + cn) ^2)))))]|.| * (- (sqrt (((p2 `2) / |.p2.|) ^2))) by A17, XCMPLX_1:76 ; hence ex x being set st ( x in dom (cn -FanMorphS) & y = (cn -FanMorphS) . x ) by A19, A21, A20, A11, EUCLID:53; ::_thesis: verum end; caseA22: ( (p2 `1) / |.p2.| < 0 & p2 `2 <= 0 & p2 <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st ( x in dom (cn -FanMorphS) & y = (cn -FanMorphS) . x ) A23: 1 + cn >= 0 by A1, XREAL_1:148; 1 - cn > 0 by A2, XREAL_1:149; then A24: (1 - cn) + cn >= (((p2 `1) / |.p2.|) * (1 + cn)) + cn by A22, A23, XREAL_1:7; A25: 1 + cn > 0 by A1, XREAL_1:148; |.p2.| <> 0 by A22, TOPRNS_1:24; then A26: |.p2.| ^2 > 0 by SQUARE_1:12; 0 <= (p2 `2) ^2 by XREAL_1:63; then ( |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) & 0 + ((p2 `1) ^2) <= ((p2 `1) ^2) + ((p2 `2) ^2) ) by JGRAPH_3:1, XREAL_1:7; then ((p2 `1) ^2) / (|.p2.| ^2) <= (|.p2.| ^2) / (|.p2.| ^2) by XREAL_1:72; then ((p2 `1) ^2) / (|.p2.| ^2) <= 1 by A26, XCMPLX_1:60; then ((p2 `1) / |.p2.|) ^2 <= 1 by XCMPLX_1:76; then (p2 `1) / |.p2.| >= - 1 by SQUARE_1:51; then ((p2 `1) / |.p2.|) * (1 + cn) >= (- 1) * (1 + cn) by A25, XREAL_1:64; then ((((p2 `1) / |.p2.|) * (1 + cn)) + cn) - cn >= (- 1) - cn ; then (((p2 `1) / |.p2.|) * (1 + cn)) + cn >= - 1 by XREAL_1:9; then 1 ^2 >= ((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2 by A24, SQUARE_1:49; then A27: 1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2) >= 0 by XREAL_1:48; then A28: sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)) >= 0 by SQUARE_1:def_2; A29: dom (cn -FanMorphS) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; set px = |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|; A30: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| `1 = |.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn) by EUCLID:52; A31: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| `2 = - (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))) by EUCLID:52; then |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2 = ((- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))))) ^2) + ((|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)) ^2) by A30, JGRAPH_3:1 .= ((|.p2.| ^2) * ((sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))) ^2)) + ((|.p2.| ^2) * (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)) ; then A32: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2 = ((|.p2.| ^2) * (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2))) + ((|.p2.| ^2) * (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)) by A27, SQUARE_1:def_2 .= |.p2.| ^2 ; then A33: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| = sqrt (|.p2.| ^2) by SQUARE_1:22 .= |.p2.| by SQUARE_1:22 ; then A34: |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| <> 0. (TOP-REAL 2) by A22, TOPRNS_1:23, TOPRNS_1:24; (((p2 `1) / |.p2.|) * (1 + cn)) + cn <= 0 + cn by A22, A23, XREAL_1:7; then (|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| <= cn by A22, A30, A33, TOPRNS_1:24, XCMPLX_1:89; then A35: (cn -FanMorphS) . |[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| = |[(|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * ((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.|) - cn) / (1 + cn))),(|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (1 - (((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.|) - cn) / (1 + cn)) ^2)))))]| by A1, A2, A31, A28, A34, Th115; A36: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (((p2 `2) / |.p2.|) ^2))) = |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt ((- ((p2 `2) / |.p2.|)) ^2))) .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (- ((p2 `2) / |.p2.|))) by A22, SQUARE_1:22 .= p2 `2 by A22, A33, TOPRNS_1:24, XCMPLX_1:87 ; A37: |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * ((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.|) - cn) / (1 + cn)) = |.p2.| * ((((((p2 `1) / |.p2.|) * (1 + cn)) + cn) - cn) / (1 + cn)) by A22, A30, A33, TOPRNS_1:24, XCMPLX_1:89 .= |.p2.| * ((p2 `1) / |.p2.|) by A25, XCMPLX_1:89 .= p2 `1 by A22, TOPRNS_1:24, XCMPLX_1:87 ; then |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (1 - (((((|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]| `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.|) - cn) / (1 + cn)) ^2)))) = |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (1 - (((p2 `1) / |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.|) ^2)))) by A22, A33, TOPRNS_1:24, XCMPLX_1:89 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (1 - (((p2 `1) ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2))))) by XCMPLX_1:76 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (((|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2)) - (((p2 `1) ^2) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2))))) by A26, A32, XCMPLX_1:60 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (((|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2) - ((p2 `1) ^2)) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2)))) by XCMPLX_1:120 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (((((p2 `1) ^2) + ((p2 `2) ^2)) - ((p2 `1) ^2)) / (|.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| ^2)))) by A32, JGRAPH_3:1 .= |.|[(|.p2.| * ((((p2 `1) / |.p2.|) * (1 + cn)) + cn)),(- (|.p2.| * (sqrt (1 - (((((p2 `1) / |.p2.|) * (1 + cn)) + cn) ^2)))))]|.| * (- (sqrt (((p2 `2) / |.p2.|) ^2))) by A33, XCMPLX_1:76 ; hence ex x being set st ( x in dom (cn -FanMorphS) & y = (cn -FanMorphS) . x ) by A35, A37, A36, A29, EUCLID:53; ::_thesis: verum end; end; end; hence y in rng f by A3, FUNCT_1:def_3; ::_thesis: verum end; hence rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) by A3, XBOOLE_0:def_10; ::_thesis: verum end; hence rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) ; ::_thesis: verum end; theorem Th135: :: JGRAPH_4:135 for cn being Real for p2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 holds ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphS) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphS) . p2 in V2 ) ) proof reconsider O = 0. (TOP-REAL 2) as Point of (Euclid 2) by EUCLID:67; let cn be Real; ::_thesis: for p2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 holds ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphS) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphS) . p2 in V2 ) ) let p2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 implies ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphS) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphS) . p2 in V2 ) ) ) A1: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; then reconsider V0 = Ball (O,(|.p2.| + 1)) as Subset of (TOP-REAL 2) ; ( O in V0 & V0 c= cl_Ball (O,(|.p2.| + 1)) ) by GOBOARD6:1, METRIC_1:14; then reconsider K0 = cl_Ball (O,(|.p2.| + 1)) as non empty compact Subset of (TOP-REAL 2) by A1, Th15; set q3 = (cn -FanMorphS) . p2; reconsider VV0 = V0 as Subset of (TopSpaceMetr (Euclid 2)) ; reconsider u2 = p2 as Point of (Euclid 2) by EUCLID:67; reconsider u3 = (cn -FanMorphS) . p2 as Point of (Euclid 2) by EUCLID:67; A2: (cn -FanMorphS) .: K0 c= K0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (cn -FanMorphS) .: K0 or y in K0 ) assume y in (cn -FanMorphS) .: K0 ; ::_thesis: y in K0 then consider x being set such that A3: x in dom (cn -FanMorphS) and A4: x in K0 and A5: y = (cn -FanMorphS) . x by FUNCT_1:def_6; reconsider q = x as Point of (TOP-REAL 2) by A3; reconsider uq = q as Point of (Euclid 2) by EUCLID:67; dist (O,uq) <= |.p2.| + 1 by A4, METRIC_1:12; then |.((0. (TOP-REAL 2)) - q).| <= |.p2.| + 1 by JGRAPH_1:28; then |.(- q).| <= |.p2.| + 1 by EUCLID:27; then A6: |.q.| <= |.p2.| + 1 by TOPRNS_1:26; A7: y in rng (cn -FanMorphS) by A3, A5, FUNCT_1:def_3; then reconsider u = y as Point of (Euclid 2) by EUCLID:67; reconsider q4 = y as Point of (TOP-REAL 2) by A7; |.q4.| = |.q.| by A5, Th128; then |.(- q4).| <= |.p2.| + 1 by A6, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - q4).| <= |.p2.| + 1 by EUCLID:27; then dist (O,u) <= |.p2.| + 1 by JGRAPH_1:28; hence y in K0 by METRIC_1:12; ::_thesis: verum end; VV0 is open by TOPMETR:14; then A8: V0 is open by Lm11, PRE_TOPC:30; A9: |.p2.| < |.p2.| + 1 by XREAL_1:29; then |.(- p2).| < |.p2.| + 1 by TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - p2).| < |.p2.| + 1 by EUCLID:27; then dist (O,u2) < |.p2.| + 1 by JGRAPH_1:28; then A10: p2 in V0 by METRIC_1:11; |.((cn -FanMorphS) . p2).| = |.p2.| by Th128; then |.(- ((cn -FanMorphS) . p2)).| < |.p2.| + 1 by A9, TOPRNS_1:26; then |.((0. (TOP-REAL 2)) - ((cn -FanMorphS) . p2)).| < |.p2.| + 1 by EUCLID:27; then dist (O,u3) < |.p2.| + 1 by JGRAPH_1:28; then A11: (cn -FanMorphS) . p2 in V0 by METRIC_1:11; assume A12: ( - 1 < cn & cn < 1 ) ; ::_thesis: ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphS) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphS) . p2 in V2 ) ) K0 c= (cn -FanMorphS) .: K0 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in K0 or y in (cn -FanMorphS) .: K0 ) assume A13: y in K0 ; ::_thesis: y in (cn -FanMorphS) .: K0 then reconsider q4 = y as Point of (TOP-REAL 2) ; reconsider y = y as Point of (Euclid 2) by A13; the carrier of (TOP-REAL 2) c= rng (cn -FanMorphS) by A12, Th134; then q4 in rng (cn -FanMorphS) by TARSKI:def_3; then consider x being set such that A14: x in dom (cn -FanMorphS) and A15: y = (cn -FanMorphS) . x by FUNCT_1:def_3; reconsider x = x as Point of (Euclid 2) by A14, Lm11; reconsider q = x as Point of (TOP-REAL 2) by A14; |.q4.| = |.q.| by A15, Th128; then q in K0 by A13, Lm12; hence y in (cn -FanMorphS) .: K0 by A14, A15, FUNCT_1:def_6; ::_thesis: verum end; then K0 = (cn -FanMorphS) .: K0 by A2, XBOOLE_0:def_10; hence ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphS) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphS) . p2 in V2 ) ) by A10, A8, A11, METRIC_1:14; ::_thesis: verum end; theorem :: JGRAPH_4:136 for cn being Real st - 1 < cn & cn < 1 holds ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = cn -FanMorphS & f is being_homeomorphism ) proof let cn be Real; ::_thesis: ( - 1 < cn & cn < 1 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = cn -FanMorphS & f is being_homeomorphism ) ) set f = cn -FanMorphS ; assume A1: ( - 1 < cn & cn < 1 ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = cn -FanMorphS & f is being_homeomorphism ) then A2: for p2 being Point of (TOP-REAL 2) ex K being non empty compact Subset of (TOP-REAL 2) st ( K = (cn -FanMorphS) .: K & ex V2 being Subset of (TOP-REAL 2) st ( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphS) . p2 in V2 ) ) by Th135; ( rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) & cn -FanMorphS is continuous ) by A1, Th132, Th134; then cn -FanMorphS is being_homeomorphism by A1, A2, Th3, Th133; hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( f = cn -FanMorphS & f is being_homeomorphism ) ; ::_thesis: verum end; theorem Th137: :: JGRAPH_4:137 for cn being Real for q being Point of (TOP-REAL 2) st cn < 1 & q `2 < 0 & (q `1) / |.q.| >= cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 >= 0 ) proof let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st cn < 1 & q `2 < 0 & (q `1) / |.q.| >= cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 >= 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( cn < 1 & q `2 < 0 & (q `1) / |.q.| >= cn implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 >= 0 ) ) assume that A1: cn < 1 and A2: q `2 < 0 and A3: (q `1) / |.q.| >= cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 >= 0 ) A4: 1 - cn > 0 by A1, XREAL_1:149; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphS) . q implies ( p `2 < 0 & p `1 >= 0 ) ) set qz = p; assume p = (cn -FanMorphS) . q ; ::_thesis: ( p `2 < 0 & p `1 >= 0 ) then A5: p = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by A2, A3, Th113; then A6: p `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52; A7: ((q `1) / |.q.|) - cn >= 0 by A3, XREAL_1:48; A8: |.q.| <> 0 by A2, JGRAPH_2:3, TOPRNS_1:24; then A9: |.q.| ^2 > 0 by SQUARE_1:12; ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) < ((q `1) ^2) + ((q `2) ^2) ) by A2, JGRAPH_3:1, SQUARE_1:12, XREAL_1:8; then ((q `1) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A9, XREAL_1:74; then ((q `1) ^2) / (|.q.| ^2) < 1 by A9, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 < 1 by XCMPLX_1:76; then 1 > (q `1) / |.q.| by SQUARE_1:52; then 1 - cn > ((q `1) / |.q.|) - cn by XREAL_1:9; then - (1 - cn) < - (((q `1) / |.q.|) - cn) by XREAL_1:24; then (- (1 - cn)) / (1 - cn) < (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A4, XREAL_1:74; then - 1 < (- (((q `1) / |.q.|) - cn)) / (1 - cn) by A4, XCMPLX_1:197; then ((- (((q `1) / |.q.|) - cn)) / (1 - cn)) ^2 < 1 ^2 by A4, A7, SQUARE_1:50; hence ( p `2 < 0 & p `1 >= 0 ) by A5, A8, A4, A6, A7, Lm13, EUCLID:52, XREAL_1:132; ::_thesis: verum end; theorem Th138: :: JGRAPH_4:138 for cn being Real for q being Point of (TOP-REAL 2) st - 1 < cn & q `2 < 0 & (q `1) / |.q.| < cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 < 0 ) proof let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < cn & q `2 < 0 & (q `1) / |.q.| < cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 < 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & q `2 < 0 & (q `1) / |.q.| < cn implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 < 0 ) ) assume that A1: - 1 < cn and A2: q `2 < 0 and A3: (q `1) / |.q.| < cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 < 0 ) A4: 1 + cn > 0 by A1, XREAL_1:148; A5: ((q `1) / |.q.|) - cn < 0 by A3, XREAL_1:49; then - (((q `1) / |.q.|) - cn) > 0 by XREAL_1:58; then (- (1 + cn)) / (1 + cn) < (- (((q `1) / |.q.|) - cn)) / (1 + cn) by A4, XREAL_1:74; then A6: - 1 < (- (((q `1) / |.q.|) - cn)) / (1 + cn) by A4, XCMPLX_1:197; A7: |.q.| <> 0 by A2, JGRAPH_2:3, TOPRNS_1:24; then A8: |.q.| ^2 > 0 by SQUARE_1:12; ( |.q.| ^2 = ((q `1) ^2) + ((q `2) ^2) & 0 + ((q `1) ^2) < ((q `1) ^2) + ((q `2) ^2) ) by A2, JGRAPH_3:1, SQUARE_1:12, XREAL_1:8; then ((q `1) ^2) / (|.q.| ^2) < (|.q.| ^2) / (|.q.| ^2) by A8, XREAL_1:74; then ((q `1) ^2) / (|.q.| ^2) < 1 by A8, XCMPLX_1:60; then ((q `1) / |.q.|) ^2 < 1 by XCMPLX_1:76; then - 1 < (q `1) / |.q.| by SQUARE_1:52; then (- 1) - cn < ((q `1) / |.q.|) - cn by XREAL_1:9; then - (- (1 + cn)) > - (((q `1) / |.q.|) - cn) by XREAL_1:24; then (- (((q `1) / |.q.|) - cn)) / (1 + cn) < 1 by A4, XREAL_1:191; then ((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2 < 1 ^2 by A6, SQUARE_1:50; then 1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2) > 0 by XREAL_1:50; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) / (1 + cn)) ^2)) > 0 by SQUARE_1:25; then sqrt (1 - (((- (((q `1) / |.q.|) - cn)) ^2) / ((1 + cn) ^2))) > 0 by XCMPLX_1:76; then sqrt (1 - (((((q `1) / |.q.|) - cn) ^2) / ((1 + cn) ^2))) > 0 ; then - (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))) > 0 by XCMPLX_1:76; then A9: - (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))) < 0 ; let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphS) . q implies ( p `2 < 0 & p `1 < 0 ) ) set qz = p; assume p = (cn -FanMorphS) . q ; ::_thesis: ( p `2 < 0 & p `1 < 0 ) then p = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| by A2, A3, Th114; then A10: ( p `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))) & p `1 = |.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn)) ) by EUCLID:52; (((q `1) / |.q.|) - cn) / (1 + cn) < 0 by A1, A5, XREAL_1:141, XREAL_1:148; hence ( p `2 < 0 & p `1 < 0 ) by A7, A10, A9, XREAL_1:132; ::_thesis: verum end; theorem Th139: :: JGRAPH_4:139 for cn being Real for q1, q2 being Point of (TOP-REAL 2) st cn < 1 & q1 `2 < 0 & (q1 `1) / |.q1.| >= cn & q2 `2 < 0 & (q2 `1) / |.q2.| >= cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| proof let cn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st cn < 1 & q1 `2 < 0 & (q1 `1) / |.q1.| >= cn & q2 `2 < 0 & (q2 `1) / |.q2.| >= cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( cn < 1 & q1 `2 < 0 & (q1 `1) / |.q1.| >= cn & q2 `2 < 0 & (q2 `1) / |.q2.| >= cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A1: cn < 1 and A2: q1 `2 < 0 and A3: (q1 `1) / |.q1.| >= cn and A4: q2 `2 < 0 and A5: (q2 `1) / |.q2.| >= cn and A6: (q1 `1) / |.q1.| < (q2 `1) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| A7: ( ((q1 `1) / |.q1.|) - cn < ((q2 `1) / |.q2.|) - cn & 1 - cn > 0 ) by A1, A6, XREAL_1:9, XREAL_1:149; let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A8: p1 = (cn -FanMorphS) . q1 and A9: p2 = (cn -FanMorphS) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| A10: |.p2.| = |.q2.| by A9, Th128; p2 = |[(|.q2.| * ((((q2 `1) / |.q2.|) - cn) / (1 - cn))),(|.q2.| * (- (sqrt (1 - (((((q2 `1) / |.q2.|) - cn) / (1 - cn)) ^2)))))]| by A4, A5, A9, Th113; then A11: p2 `1 = |.q2.| * ((((q2 `1) / |.q2.|) - cn) / (1 - cn)) by EUCLID:52; |.q2.| > 0 by A4, Lm1, JGRAPH_2:3; then A12: (p2 `1) / |.p2.| = (((q2 `1) / |.q2.|) - cn) / (1 - cn) by A11, A10, XCMPLX_1:89; p1 = |[(|.q1.| * ((((q1 `1) / |.q1.|) - cn) / (1 - cn))),(|.q1.| * (- (sqrt (1 - (((((q1 `1) / |.q1.|) - cn) / (1 - cn)) ^2)))))]| by A2, A3, A8, Th113; then A13: p1 `1 = |.q1.| * ((((q1 `1) / |.q1.|) - cn) / (1 - cn)) by EUCLID:52; A14: |.p1.| = |.q1.| by A8, Th128; |.q1.| > 0 by A2, Lm1, JGRAPH_2:3; then (p1 `1) / |.p1.| = (((q1 `1) / |.q1.|) - cn) / (1 - cn) by A13, A14, XCMPLX_1:89; hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A12, A7, XREAL_1:74; ::_thesis: verum end; theorem Th140: :: JGRAPH_4:140 for cn being Real for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & q1 `2 < 0 & (q1 `1) / |.q1.| < cn & q2 `2 < 0 & (q2 `1) / |.q2.| < cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| proof let cn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & q1 `2 < 0 & (q1 `1) / |.q1.| < cn & q2 `2 < 0 & (q2 `1) / |.q2.| < cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & q1 `2 < 0 & (q1 `1) / |.q1.| < cn & q2 `2 < 0 & (q2 `1) / |.q2.| < cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A1: - 1 < cn and A2: q1 `2 < 0 and A3: (q1 `1) / |.q1.| < cn and A4: q2 `2 < 0 and A5: (q2 `1) / |.q2.| < cn and A6: (q1 `1) / |.q1.| < (q2 `1) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| A7: ( ((q1 `1) / |.q1.|) - cn < ((q2 `1) / |.q2.|) - cn & 1 + cn > 0 ) by A1, A6, XREAL_1:9, XREAL_1:148; let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A8: p1 = (cn -FanMorphS) . q1 and A9: p2 = (cn -FanMorphS) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| A10: |.p2.| = |.q2.| by A9, Th128; p2 = |[(|.q2.| * ((((q2 `1) / |.q2.|) - cn) / (1 + cn))),(|.q2.| * (- (sqrt (1 - (((((q2 `1) / |.q2.|) - cn) / (1 + cn)) ^2)))))]| by A4, A5, A9, Th114; then A11: p2 `1 = |.q2.| * ((((q2 `1) / |.q2.|) - cn) / (1 + cn)) by EUCLID:52; |.q2.| > 0 by A4, Lm1, JGRAPH_2:3; then A12: (p2 `1) / |.p2.| = (((q2 `1) / |.q2.|) - cn) / (1 + cn) by A11, A10, XCMPLX_1:89; p1 = |[(|.q1.| * ((((q1 `1) / |.q1.|) - cn) / (1 + cn))),(|.q1.| * (- (sqrt (1 - (((((q1 `1) / |.q1.|) - cn) / (1 + cn)) ^2)))))]| by A2, A3, A8, Th114; then A13: p1 `1 = |.q1.| * ((((q1 `1) / |.q1.|) - cn) / (1 + cn)) by EUCLID:52; A14: |.p1.| = |.q1.| by A8, Th128; |.q1.| > 0 by A2, Lm1, JGRAPH_2:3; then (p1 `1) / |.p1.| = (((q1 `1) / |.q1.|) - cn) / (1 + cn) by A13, A14, XCMPLX_1:89; hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A12, A7, XREAL_1:74; ::_thesis: verum end; theorem :: JGRAPH_4:141 for cn being Real for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 < 0 & q2 `2 < 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| proof let cn be Real; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 < 0 & q2 `2 < 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q1 `2 < 0 & q2 `2 < 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| implies for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A1: - 1 < cn and A2: cn < 1 and A3: q1 `2 < 0 and A4: q2 `2 < 0 and A5: (q1 `1) / |.q1.| < (q2 `1) / |.q2.| ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds (p1 `1) / |.p1.| < (p2 `1) / |.p2.| let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| ) assume that A6: p1 = (cn -FanMorphS) . q1 and A7: p2 = (cn -FanMorphS) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| percases ( ( (q1 `1) / |.q1.| >= cn & (q2 `1) / |.q2.| >= cn ) or ( (q1 `1) / |.q1.| >= cn & (q2 `1) / |.q2.| < cn ) or ( (q1 `1) / |.q1.| < cn & (q2 `1) / |.q2.| >= cn ) or ( (q1 `1) / |.q1.| < cn & (q2 `1) / |.q2.| < cn ) ) ; suppose ( (q1 `1) / |.q1.| >= cn & (q2 `1) / |.q2.| >= cn ) ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A2, A3, A4, A5, A6, A7, Th139; ::_thesis: verum end; suppose ( (q1 `1) / |.q1.| >= cn & (q2 `1) / |.q2.| < cn ) ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A5, XXREAL_0:2; ::_thesis: verum end; supposeA8: ( (q1 `1) / |.q1.| < cn & (q2 `1) / |.q2.| >= cn ) ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| then p2 `1 >= 0 by A2, A4, A7, Th137; then A9: (p2 `1) / |.p2.| >= 0 ; p1 `1 < 0 by A1, A3, A6, A8, Th138; hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A9, Lm1, JGRAPH_2:3, XREAL_1:141; ::_thesis: verum end; suppose ( (q1 `1) / |.q1.| < cn & (q2 `1) / |.q2.| < cn ) ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.| hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A1, A3, A4, A5, A6, A7, Th140; ::_thesis: verum end; end; end; theorem :: JGRAPH_4:142 for cn being Real for q being Point of (TOP-REAL 2) st q `2 < 0 & (q `1) / |.q.| = cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 = 0 ) proof let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st q `2 < 0 & (q `1) / |.q.| = cn holds for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 = 0 ) let q be Point of (TOP-REAL 2); ::_thesis: ( q `2 < 0 & (q `1) / |.q.| = cn implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 = 0 ) ) assume that A1: q `2 < 0 and A2: (q `1) / |.q.| = cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds ( p `2 < 0 & p `1 = 0 ) let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphS) . q implies ( p `2 < 0 & p `1 = 0 ) ) A3: |.q.| <> 0 by A1, JGRAPH_2:3, TOPRNS_1:24; assume p = (cn -FanMorphS) . q ; ::_thesis: ( p `2 < 0 & p `1 = 0 ) then A4: p = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| by A1, A2, Th113; then p `2 = |.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))) by EUCLID:52; hence ( p `2 < 0 & p `1 = 0 ) by A2, A4, A3, Lm13, EUCLID:52, XREAL_1:132; ::_thesis: verum end; theorem :: JGRAPH_4:143 for a being real number holds 0. (TOP-REAL 2) = (a -FanMorphS) . (0. (TOP-REAL 2)) by Th113, JGRAPH_2:3;