:: On the Simple Closed Curve Property of the Circle and the Fashoda Meet Theorem for It :: by Yatsuka Nakamura :: :: Received August 20, 2001 :: Copyright (c) 2001-2012 Association of Mizar Users begin Lm1: for x being real number holds (x ^2) + 1 > 0 proofend; Lm2: dom proj1 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; Lm3: dom proj2 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; theorem :: JGRAPH_3:1 for p being Point of (TOP-REAL 2) holds ( |.p.| = sqrt (((p `1) ^2) + ((p `2) ^2)) & |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) ) by JGRAPH_1:29, JGRAPH_1:30; theorem :: JGRAPH_3:2 for f being Function for B, C being set holds (f | B) .: C = f .: (C /\ B) proofend; theorem Th3: :: JGRAPH_3:3 for X, Y being non empty TopSpace for p0 being Point of X for D being non empty Subset of X for E being non empty Subset of Y for f being Function of X,Y st D ` = {p0} & E ` = {(f . p0)} & X is T_2 & Y is T_2 & ( for p being Point of (X | D) holds f . p <> f . p0 ) & f | D is continuous Function of (X | D),(Y | E) & ( for V being Subset of Y st f . p0 in V & V is open holds ex W being Subset of X st ( p0 in W & W is open & f .: W c= V ) ) holds f is continuous proofend; begin definition func Sq_Circ -> Function of the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2) means :Def1: :: JGRAPH_3:def 1 for p being Point of (TOP-REAL 2) holds ( ( p = 0. (TOP-REAL 2) implies it . p = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies it . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or not p <> 0. (TOP-REAL 2) or it . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ); existence ex b1 being Function of the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2) st for p being Point of (TOP-REAL 2) holds ( ( p = 0. (TOP-REAL 2) implies b1 . p = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies b1 . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or not p <> 0. (TOP-REAL 2) or b1 . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) proofend; uniqueness for b1, b2 being Function of the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2) st ( for p being Point of (TOP-REAL 2) holds ( ( p = 0. (TOP-REAL 2) implies b1 . p = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies b1 . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or not p <> 0. (TOP-REAL 2) or b1 . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) ) & ( for p being Point of (TOP-REAL 2) holds ( ( p = 0. (TOP-REAL 2) implies b2 . p = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies b2 . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or not p <> 0. (TOP-REAL 2) or b2 . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) ) holds b1 = b2 proofend; end; :: deftheorem Def1 defines Sq_Circ JGRAPH_3:def_1_:_ for b1 being Function of the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2) holds ( b1 = Sq_Circ iff for p being Point of (TOP-REAL 2) holds ( ( p = 0. (TOP-REAL 2) implies b1 . p = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies b1 . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or not p <> 0. (TOP-REAL 2) or b1 . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) ); theorem Th4: :: JGRAPH_3:4 for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) implies Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| ) ) proofend; theorem Th5: :: JGRAPH_3:5 for X being non empty TopSpace for f1 being Function of X,R^1 st f1 is continuous & ( for q being Point of X ex r being real number st ( f1 . q = r & r >= 0 ) ) 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 = sqrt r1 ) & g is continuous ) proofend; theorem Th6: :: JGRAPH_3:6 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( 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) ^2 ) & g is continuous ) proofend; theorem Th7: :: JGRAPH_3:7 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( 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 = 1 + ((r1 / r2) ^2) ) & g is continuous ) proofend; theorem Th8: :: JGRAPH_3:8 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( 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 = sqrt (1 + ((r1 / r2) ^2)) ) & g is continuous ) proofend; theorem Th9: :: JGRAPH_3:9 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( 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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) proofend; theorem Th10: :: JGRAPH_3:10 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( 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 (1 + ((r1 / r2) ^2))) ) & g is continuous ) proofend; Lm4: for K1 being non empty Subset of (TOP-REAL 2) for q being Point of ((TOP-REAL 2) | K1) holds (proj2 | K1) . q = proj2 . q proofend; Lm5: for K1 being non empty Subset of (TOP-REAL 2) holds proj2 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1 proofend; Lm6: for K1 being non empty Subset of (TOP-REAL 2) for q being Point of ((TOP-REAL 2) | K1) holds (proj1 | K1) . q = proj1 . q proofend; Lm7: for K1 being non empty Subset of (TOP-REAL 2) holds proj1 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1 proofend; theorem Th11: :: JGRAPH_3:11 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) holds f is continuous proofend; theorem Th12: :: JGRAPH_3:12 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) holds f is continuous proofend; theorem Th13: :: JGRAPH_3:13 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) holds f is continuous proofend; theorem Th14: :: JGRAPH_3:14 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) holds f is continuous proofend; Lm8: 0.REAL 2 = 0. (TOP-REAL 2) by EUCLID:66; Lm9: ( ( (1.REAL 2) `2 <= (1.REAL 2) `1 & - ((1.REAL 2) `1) <= (1.REAL 2) `2 ) or ( (1.REAL 2) `2 >= (1.REAL 2) `1 & (1.REAL 2) `2 <= - ((1.REAL 2) `1) ) ) by JGRAPH_2:5; Lm10: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm8, REVROT_1:19; Lm11: for K1 being non empty Subset of (TOP-REAL 2) holds dom (proj2 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1) proofend; Lm12: for K1 being non empty Subset of (TOP-REAL 2) holds dom (proj1 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1) proofend; Lm13: NonZero (TOP-REAL 2) <> {} by JGRAPH_2:9; theorem Th15: :: JGRAPH_3:15 for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Sq_Circ | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds f is continuous proofend; Lm14: ( ( (1.REAL 2) `1 <= (1.REAL 2) `2 & - ((1.REAL 2) `2) <= (1.REAL 2) `1 ) or ( (1.REAL 2) `1 >= (1.REAL 2) `2 & (1.REAL 2) `1 <= - ((1.REAL 2) `2) ) ) by JGRAPH_2:5; Lm15: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm8, REVROT_1:19; theorem Th16: :: JGRAPH_3:16 for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Sq_Circ | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds f is continuous proofend; scheme :: JGRAPH_3:sch 1 TopIncl{ P1[ set ] } : { p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) ) } c= NonZero (TOP-REAL 2) proofend; scheme :: JGRAPH_3:sch 2 TopInter{ P1[ set ] } : { p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) ) } = { p7 where p7 is Point of (TOP-REAL 2) : P1[p7] } /\ (NonZero (TOP-REAL 2)) proofend; theorem Th17: :: JGRAPH_3:17 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 f = Sq_Circ | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) proofend; theorem Th18: :: JGRAPH_3:18 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 f = Sq_Circ | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) proofend; theorem Th19: :: JGRAPH_3:19 for D being non empty Subset of (TOP-REAL 2) st D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = Sq_Circ | D & h is continuous ) proofend; theorem Th20: :: JGRAPH_3:20 for D being non empty Subset of (TOP-REAL 2) st D = NonZero (TOP-REAL 2) holds D ` = {(0. (TOP-REAL 2))} proofend; Lm16: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; theorem Th21: :: JGRAPH_3:21 ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = Sq_Circ & h is continuous ) proofend; theorem Th22: :: JGRAPH_3:22 Sq_Circ is one-to-one proofend; registration cluster Sq_Circ -> one-to-one ; coherence Sq_Circ is one-to-one by Th22; end; theorem Th23: :: JGRAPH_3:23 for Kb, Cb being Subset of (TOP-REAL 2) st Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| = 1 } holds Sq_Circ .: Kb = Cb proofend; theorem Th24: :: JGRAPH_3:24 for P, Kb being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | Kb),((TOP-REAL 2) | P) st Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } & f is being_homeomorphism holds P is being_simple_closed_curve proofend; theorem Th25: :: JGRAPH_3:25 for Kb being Subset of (TOP-REAL 2) st Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } holds ( Kb is being_simple_closed_curve & Kb is compact ) proofend; theorem :: JGRAPH_3:26 for Cb being Subset of (TOP-REAL 2) st Cb = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } holds Cb is being_simple_closed_curve proofend; begin theorem :: JGRAPH_3:27 for K0, C0 being Subset of (TOP-REAL 2) st K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 ) } & C0 = { p1 where p1 is Point of (TOP-REAL 2) : |.p1.| <= 1 } holds Sq_Circ " C0 c= K0 proofend; theorem Th28: :: JGRAPH_3:28 for p being Point of (TOP-REAL 2) holds ( ( p = 0. (TOP-REAL 2) implies (Sq_Circ ") . p = 0. (TOP-REAL 2) ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) proofend; theorem Th29: :: JGRAPH_3:29 Sq_Circ " is Function of (TOP-REAL 2),(TOP-REAL 2) proofend; theorem Th30: :: JGRAPH_3:30 for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ( ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) implies (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| ) ) proofend; theorem Th31: :: JGRAPH_3:31 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( 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 * (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) proofend; theorem Th32: :: JGRAPH_3:32 for X being non empty TopSpace for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( 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 (1 + ((r1 / r2) ^2))) ) & g is continuous ) proofend; theorem Th33: :: JGRAPH_3:33 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) holds f is continuous proofend; theorem Th34: :: JGRAPH_3:34 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `1 <> 0 ) holds f is continuous proofend; theorem Th35: :: JGRAPH_3:35 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) holds f is continuous proofend; theorem Th36: :: JGRAPH_3:36 for K1 being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds f . p = (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds q `2 <> 0 ) holds f is continuous proofend; Lm17: for K1 being non empty Subset of (TOP-REAL 2) holds proj2 * ((Sq_Circ ") | K1) is Function of ((TOP-REAL 2) | K1),R^1 proofend; Lm18: for K1 being non empty Subset of (TOP-REAL 2) holds proj1 * ((Sq_Circ ") | K1) is Function of ((TOP-REAL 2) | K1),R^1 proofend; theorem Th37: :: JGRAPH_3:37 for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = (Sq_Circ ") | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds f is continuous proofend; theorem Th38: :: JGRAPH_3:38 for K0, B0 being Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = (Sq_Circ ") | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds f is continuous proofend; theorem Th39: :: JGRAPH_3:39 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 f = (Sq_Circ ") | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) proofend; theorem Th40: :: JGRAPH_3:40 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 f = (Sq_Circ ") | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds ( f is continuous & K0 is closed ) proofend; theorem Th41: :: JGRAPH_3:41 for D being non empty Subset of (TOP-REAL 2) st D ` = {(0. (TOP-REAL 2))} holds ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st ( h = (Sq_Circ ") | D & h is continuous ) proofend; theorem Th42: :: JGRAPH_3:42 ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st ( h = Sq_Circ " & h is continuous ) proofend; theorem Th43: :: JGRAPH_3:43 ( Sq_Circ is Function of (TOP-REAL 2),(TOP-REAL 2) & rng Sq_Circ = the carrier of (TOP-REAL 2) & ( for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds f is being_homeomorphism ) ) proofend; Lm19: now__::_thesis:_for_pz2,_pz1_being_real_number_st_(((pz2_^2)_+_(pz1_^2))_-_1)_*_(pz2_^2)_<=_pz1_^2_holds_ ((pz2_^2)_-_1)_*_((pz2_^2)_+_(pz1_^2))_<=_0 let pz2, pz1 be real number ; ::_thesis: ( (((pz2 ^2) + (pz1 ^2)) - 1) * (pz2 ^2) <= pz1 ^2 implies ((pz2 ^2) - 1) * ((pz2 ^2) + (pz1 ^2)) <= 0 ) assume (((pz2 ^2) + (pz1 ^2)) - 1) * (pz2 ^2) <= pz1 ^2 ; ::_thesis: ((pz2 ^2) - 1) * ((pz2 ^2) + (pz1 ^2)) <= 0 then (((pz2 ^2) * (pz2 ^2)) + ((pz2 ^2) * ((pz1 ^2) - 1))) - (pz1 ^2) <= (pz1 ^2) - (pz1 ^2) by XREAL_1:9; hence ((pz2 ^2) - 1) * ((pz2 ^2) + (pz1 ^2)) <= 0 ; ::_thesis: verum end; Lm20: now__::_thesis:_for_px1_being_real_number_holds_ (_not_(px1_^2)_-_1_=_0_or_px1_=_1_or_px1_=_-_1_) let px1 be real number ; ::_thesis: ( not (px1 ^2) - 1 = 0 or px1 = 1 or px1 = - 1 ) assume (px1 ^2) - 1 = 0 ; ::_thesis: ( px1 = 1 or px1 = - 1 ) then (px1 - 1) * (px1 + 1) = 0 ; then ( px1 - 1 = 0 or px1 + 1 = 0 ) by XCMPLX_1:6; hence ( px1 = 1 or px1 = - 1 ) ; ::_thesis: verum end; theorem :: JGRAPH_3:44 for f, g being Function of I[01],(TOP-REAL 2) for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2) for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 holds rng f meets rng g proofend;