:: JGRAPH_3 semantic presentation

begin

 Lm1:  for x being real number holds (x ^2) + 1 > 0
proof
let x be real number ; ::_thesis: (x ^2) + 1 > 0
 x ^2 >= 0 by XREAL_1:63;
hence  (x ^2) + 1 > 0 ; ::_thesis: verum
end;

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)
proof
let f be Function; ::_thesis: for B, C being set holds (f | B) .: C = f .: (C /\ B)
let B, C be set ; ::_thesis: (f | B) .: C = f .: (C /\ B)
thus  (f | B) .: C c= f .: (C /\ B) :: according to XBOOLE_0:def_10 ::_thesis: f .: (C /\ B) c= (f | B) .: C
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (f | B) .: C or x in f .: (C /\ B) )
assume  x in (f | B) .: C ; ::_thesis: x in f .: (C /\ B)
then consider y being set  such that
A1: y in dom (f | B) and
A2: y in C and
A3: x = (f | B) . y by FUNCT_1:def_6;
A4: (f | B) . y = f . y by A1, FUNCT_1:47;
A5: dom (f | B) = (dom f) /\ B by RELAT_1:61;
then  y in B by A1, XBOOLE_0:def_4;
then A6: y in C /\ B by A2, XBOOLE_0:def_4;
 y in dom f by A1, A5, XBOOLE_0:def_4;
hence  x in f .: (C /\ B) by A3, A6, A4, FUNCT_1:def_6; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f .: (C /\ B) or x in (f | B) .: C )
assume  x in f .: (C /\ B) ; ::_thesis: x in (f | B) .: C
then consider y being set  such that
A7: y in dom f and
A8: y in C /\ B and
A9: x = f . y by FUNCT_1:def_6;
A10: y in C by A8, XBOOLE_0:def_4;
 y in B by A8, XBOOLE_0:def_4;
then  y in (dom f) /\ B by A7, XBOOLE_0:def_4;
then A11: y in dom (f | B) by RELAT_1:61;
then  (f | B) . y = f . y by FUNCT_1:47;
hence  x in (f | B) .: C by A9, A10, A11, FUNCT_1:def_6; ::_thesis: verum
end;

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
proof
let X, Y be non empty TopSpace; ::_thesis: 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
let p0 be Point of X; ::_thesis: 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
let D be non empty Subset of X; ::_thesis: 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
let E be non empty Subset of Y; ::_thesis: 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
let f be Function of X,Y; ::_thesis: ( 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 ) ) implies f is continuous )
assume that
A1: D ` = {p0} and
A2: E ` = {(f . p0)} and
A3: X is T_2 and
A4: Y is T_2 and
A5: for p being Point of (X | D) holds f . p <> f . p0 and
A6: f | D is continuous Function of (X | D),(Y | E) and
A7: 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 ) ; ::_thesis: f is continuous
 for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V )
proof
A8: the carrier of (X | D) = D by PRE_TOPC:8;
let p be Point of X; ::_thesis: for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V )
let V be Subset of Y; ::_thesis: ( f . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & f .: W c= V ) )
assume that
A9: f . p in V and
A10: V is open ; ::_thesis: ex W being Subset of X st
( p in W & W is open & f .: W c= V )
percases ( p = p0 or p <> p0 ) ;
suppose p = p0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & f .: W c= V )
hence  ex W being Subset of X st
( p in W & W is open & f .: W c= V ) by A7, A9, A10; ::_thesis: verum
end;
supposeA11: p <> p0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & f .: W c= V )
then  not p in D ` by A1, TARSKI:def_1;
then  p in the carrier of X \ (D `) by XBOOLE_0:def_5;
then A12: p in (D `) ` by SUBSET_1:def_4;
then  f . p <> f . p0 by A5, A8;
then consider G1, G2 being Subset of Y such that
A13: G1 is open and
 G2 is open and
A14: f . p in G1 and
 f . p0 in G2 and
 G1 misses G2 by A4, PRE_TOPC:def_10;
A15: [#] (X | D) = D by PRE_TOPC:def_5;
then reconsider p22 = p as Point of (X | D) by A12;
consider h being Function of (X | D),(Y | E) such that
A16: h = f | D and
A17: h is continuous by A6;
A18: h . p = f . p by A12, A16, FUNCT_1:49;
A19: [#] (Y | E) = E by PRE_TOPC:def_5;
then reconsider V20 = (G1 /\ V) /\ E as Subset of (Y | E) by XBOOLE_1:17;
 G1 /\ V is open by A10, A13, TOPS_1:11;
then A20: V20 is open by A19, TOPS_2:24;
 f . p <> f . p0 by A5, A12, A15;
then  not f . p in E ` by A2, TARSKI:def_1;
then  not f . p in the carrier of Y \ E by SUBSET_1:def_4;
then A21: h . p22 in E by A18, XBOOLE_0:def_5;
 h . p22 in G1 /\ V by A9, A14, A18, XBOOLE_0:def_4;
then  h . p22 in V20 by A21, XBOOLE_0:def_4;
then consider W2 being Subset of (X | D) such that
A22: p22 in W2 and
A23: W2 is open and
A24: h .: W2 c= V20 by A17, A20, JGRAPH_2:10;
consider W3b being Subset of X such that
A25: W3b is open and
A26: W2 = W3b /\ ([#] (X | D)) by A23, TOPS_2:24;
consider H1, H2 being Subset of X such that
A27: H1 is open and
 H2 is open and
A28: p in H1 and
A29: p0 in H2 and
A30: H1 misses H2 by A3, A11, PRE_TOPC:def_10;
 p22 in W3b by A22, A26, XBOOLE_0:def_4;
then A31: p in H1 /\ W3b by A28, XBOOLE_0:def_4;
reconsider W3 = H1 /\ W3b as Subset of X ;
A32: W3 c= W3b by XBOOLE_1:17;
A33: f .: W3 c= h .: W2
proof
let xx be set ; :: according to TARSKI:def_3 ::_thesis: ( not xx in f .: W3 or xx in h .: W2 )
assume  xx in f .: W3 ; ::_thesis: xx in h .: W2
then consider yy being set  such that
A34: yy in dom f and
A35: yy in W3 and
A36: xx = f . yy by FUNCT_1:def_6;
 H2 c= H1 ` by A30, SUBSET_1:23;
then  D ` c= H1 ` by A1, A29, ZFMISC_1:31;
then  ( W3 c= H1 & H1 c= D ) by SUBSET_1:12, XBOOLE_1:17;
then A37: W3 c= D by XBOOLE_1:1;
then A38: yy in W2 by A15, A26, A32, A35, XBOOLE_0:def_4;
 dom h = (dom f) /\ D by A16, RELAT_1:61;
then A39: yy in dom h by A34, A35, A37, XBOOLE_0:def_4;
then  h . yy = f . yy by A16, FUNCT_1:47;
hence  xx in h .: W2 by A36, A39, A38, FUNCT_1:def_6; ::_thesis: verum
end;
 (G1 /\ V) /\ E c= G1 /\ V by XBOOLE_1:17;
then  ( G1 /\ V c= V & h .: W2 c= G1 /\ V ) by A24, XBOOLE_1:1, XBOOLE_1:17;
then A40: h .: W2 c= V by XBOOLE_1:1;
 H1 /\ W3b is open by A25, A27, TOPS_1:11;
hence  ex W being Subset of X st
( p in W & W is open & f .: W c= V ) by A31, A33, A40, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence  f is continuous by JGRAPH_2:10; ::_thesis: verum
end;

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))))]| ) )
proof
defpred S1[ set , set ] means  for p being Point of (TOP-REAL 2) st p = $1 holds
( ( p = 0. (TOP-REAL 2) implies $2 = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies $2 = |[((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 $2 = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) );
set BP = the carrier of (TOP-REAL 2);
A1: for x being Element of the carrier of (TOP-REAL 2) ex y being Element of the carrier of (TOP-REAL 2) st S1[x,y]
proof
let x be Element of the carrier of (TOP-REAL 2); ::_thesis: ex y being Element of the carrier of (TOP-REAL 2) st S1[x,y]
set q = x;
percases ( x = 0. (TOP-REAL 2) or ( ( ( x `2 <= x `1 & - (x `1) <= x `2 ) or ( x `2 >= x `1 & x `2 <= - (x `1) ) ) & x <> 0. (TOP-REAL 2) ) or ( not ( x `2 <= x `1 & - (x `1) <= x `2 ) & not ( x `2 >= x `1 & x `2 <= - (x `1) ) & x <> 0. (TOP-REAL 2) ) ) ;
suppose x = 0. (TOP-REAL 2) ; ::_thesis: ex y being Element of the carrier of (TOP-REAL 2) st S1[x,y]
then  for p being Point of (TOP-REAL 2) st p = x holds
( ( p = 0. (TOP-REAL 2) implies 0. (TOP-REAL 2) = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies 0. (TOP-REAL 2) = |[((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 0. (TOP-REAL 2) = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) ;
hence  ex y being Element of the carrier of (TOP-REAL 2) st S1[x,y] ; ::_thesis: verum
end;
supposeA2: ( ( ( x `2 <= x `1 & - (x `1) <= x `2 ) or ( x `2 >= x `1 & x `2 <= - (x `1) ) ) & x <> 0. (TOP-REAL 2) ) ; ::_thesis: ex y being Element of the carrier of (TOP-REAL 2) st S1[x,y]
set r = |[((x `1) / (sqrt (1 + (((x `2) / (x `1)) ^2)))),((x `2) / (sqrt (1 + (((x `2) / (x `1)) ^2))))]|;
 for p being Point of (TOP-REAL 2) st p = x holds
( ( p = 0. (TOP-REAL 2) implies |[((x `1) / (sqrt (1 + (((x `2) / (x `1)) ^2)))),((x `2) / (sqrt (1 + (((x `2) / (x `1)) ^2))))]| = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies |[((x `1) / (sqrt (1 + (((x `2) / (x `1)) ^2)))),((x `2) / (sqrt (1 + (((x `2) / (x `1)) ^2))))]| = |[((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 |[((x `1) / (sqrt (1 + (((x `2) / (x `1)) ^2)))),((x `2) / (sqrt (1 + (((x `2) / (x `1)) ^2))))]| = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) by A2;
hence  ex y being Element of the carrier of (TOP-REAL 2) st S1[x,y] ; ::_thesis: verum
end;
supposeA3: ( not ( x `2 <= x `1 & - (x `1) <= x `2 ) & not ( x `2 >= x `1 & x `2 <= - (x `1) ) & x <> 0. (TOP-REAL 2) ) ; ::_thesis: ex y being Element of the carrier of (TOP-REAL 2) st S1[x,y]
set r = |[((x `1) / (sqrt (1 + (((x `1) / (x `2)) ^2)))),((x `2) / (sqrt (1 + (((x `1) / (x `2)) ^2))))]|;
 for p being Point of (TOP-REAL 2) st p = x holds
( ( p = 0. (TOP-REAL 2) implies |[((x `1) / (sqrt (1 + (((x `1) / (x `2)) ^2)))),((x `2) / (sqrt (1 + (((x `1) / (x `2)) ^2))))]| = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies |[((x `1) / (sqrt (1 + (((x `1) / (x `2)) ^2)))),((x `2) / (sqrt (1 + (((x `1) / (x `2)) ^2))))]| = |[((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 |[((x `1) / (sqrt (1 + (((x `1) / (x `2)) ^2)))),((x `2) / (sqrt (1 + (((x `1) / (x `2)) ^2))))]| = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) by A3;
hence  ex y being Element of the carrier of (TOP-REAL 2) st S1[x,y] ; ::_thesis: verum
end;
end;
end;
 ex h being Function of the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2) st
for x being Element of the carrier of (TOP-REAL 2) holds S1[x,h . x] from FUNCT_2:sch_3(A1);
then consider h being Function of the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2) such that
A4: for x being Element of the carrier of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p = x holds
( ( p = 0. (TOP-REAL 2) implies h . x = p ) & ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) implies h . x = |[((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 h . x = |[((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 h . 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 h . 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 h . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) by A4;
hence  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))))]| ) ) ; ::_thesis: verum
end;
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
proof
let h1, h2 be Function of the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2); ::_thesis: ( ( for p being Point of (TOP-REAL 2) holds
( ( p = 0. (TOP-REAL 2) implies h1 . 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 h1 . 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 h1 . 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 h2 . 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 h2 . 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 h2 . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) ) implies h1 = h2 )
assume that
A5: for p being Point of (TOP-REAL 2) holds
( ( p = 0. (TOP-REAL 2) implies h1 . 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 h1 . 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 h1 . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) and
A6: for p being Point of (TOP-REAL 2) holds
( ( p = 0. (TOP-REAL 2) implies h2 . 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 h2 . 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 h2 . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ) ) ; ::_thesis: h1 = h2
 for x being set st x in the carrier of (TOP-REAL 2) holds
h1 . x = h2 . x
proof
let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL 2) implies h1 . x = h2 . x )
assume  x in the carrier of (TOP-REAL 2) ; ::_thesis: h1 . x = h2 . x
then reconsider q = x as Point of (TOP-REAL 2) ;
percases ( q = 0. (TOP-REAL 2) or ( ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) & q <> 0. (TOP-REAL 2) ) or ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) & q <> 0. (TOP-REAL 2) ) ) ;
supposeA7: q = 0. (TOP-REAL 2) ; ::_thesis: h1 . x = h2 . x
then  h1 . q = q by A5;
hence  h1 . x = h2 . x by A6, A7; ::_thesis: verum
end;
supposeA8: ( ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) & q <> 0. (TOP-REAL 2) ) ; ::_thesis: h1 . x = h2 . x
then  h1 . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by A5;
hence  h1 . x = h2 . x by A6, A8; ::_thesis: verum
end;
supposeA9: ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) & q <> 0. (TOP-REAL 2) ) ; ::_thesis: h1 . x = h2 . x
then  h1 . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by A5;
hence  h1 . x = h2 . x by A6, A9; ::_thesis: verum
end;
end;
end;
hence  h1 = h2 by FUNCT_2:12; ::_thesis: verum
end;
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))))]| ) )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p <> 0. (TOP-REAL 2) implies ( ( ( ( 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))))]| ) ) )
A1: ( - (p `2) < p `1 implies - (- (p `2)) > - (p `1) ) by XREAL_1:24;
assume A2: p <> 0. (TOP-REAL 2) ; ::_thesis: ( ( ( ( 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))))]| ) )
hereby  ::_thesis: ( ( 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))))]| )
assume A3: ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
now__::_thesis:_(_(_p_`1_<=_p_`2_&_-_(p_`2)_<=_p_`1_&_Sq_Circ_._p_=_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_or_(_p_`1_>=_p_`2_&_p_`1_<=_-_(p_`2)_&_Sq_Circ_._p_=_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_)
percases ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by A3;
caseA4: ( p `1 <= p `2 & - (p `2) <= p `1 ) ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
now__::_thesis:_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_implies_Sq_Circ_._p_=_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)
assume A5: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
A6: now__::_thesis:_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_)
percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A5;
case ( p `2 <= p `1 & - (p `1) <= p `2 ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
hence  ( p `1 = p `2 or p `1 = - (p `2) ) by A4, XXREAL_0:1; ::_thesis: verum
end;
case ( p `2 >= p `1 & p `2 <= - (p `1) ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
then  - (p `2) >= - (- (p `1)) by XREAL_1:24;
hence  ( p `1 = p `2 or p `1 = - (p `2) ) by A4, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_p_`1_=_p_`2_&_Sq_Circ_._p_=_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_or_(_p_`1_=_-_(p_`2)_&_Sq_Circ_._p_=_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_)
percases ( p `1 = p `2 or p `1 = - (p `2) ) by A6;
case p `1 = p `2 ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, A5, Def1; ::_thesis: verum
end;
caseA7: p `1 = - (p `2) ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
then  ( p `1 <> 0 & - (p `1) = p `2 ) by A2, EUCLID:53, EUCLID:54;
then A8: (p `2) / (p `1) = - 1 by XCMPLX_1:197;
 p `2 <> 0 by A2, A7, EUCLID:53, EUCLID:54;
then  (p `1) / (p `2) = - 1 by A7, XCMPLX_1:197;
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, A5, A8, Def1; ::_thesis: verum
end;
end;
end;
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ; ::_thesis: verum
end;
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, Def1; ::_thesis: verum
end;
caseA9: ( p `1 >= p `2 & p `1 <= - (p `2) ) ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
now__::_thesis:_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_implies_Sq_Circ_._p_=_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)
assume A10: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
A11: now__::_thesis:_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_)
percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A10;
case ( p `2 <= p `1 & - (p `1) <= p `2 ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
then  - (- (p `1)) >= - (p `2) by XREAL_1:24;
hence  ( p `1 = p `2 or p `1 = - (p `2) ) by A9, XXREAL_0:1; ::_thesis: verum
end;
case ( p `2 >= p `1 & p `2 <= - (p `1) ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
hence  ( p `1 = p `2 or p `1 = - (p `2) ) by A9, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_p_`1_=_p_`2_&_Sq_Circ_._p_=_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_or_(_p_`1_=_-_(p_`2)_&_Sq_Circ_._p_=_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_)
percases ( p `1 = p `2 or p `1 = - (p `2) ) by A11;
case p `1 = p `2 ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, A10, Def1; ::_thesis: verum
end;
caseA12: p `1 = - (p `2) ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
then  ( p `1 <> 0 & - (p `1) = p `2 ) by A2, EUCLID:53, EUCLID:54;
then A13: (p `2) / (p `1) = - 1 by XCMPLX_1:197;
 p `2 <> 0 by A2, A12, EUCLID:53, EUCLID:54;
then  (p `1) / (p `2) = - 1 by A12, XCMPLX_1:197;
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, A10, A13, Def1; ::_thesis: verum
end;
end;
end;
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ; ::_thesis: verum
end;
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, Def1; ::_thesis: verum
end;
end;
end;
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ; ::_thesis: verum
end;
A14: ( - (p `2) > p `1 implies - (- (p `2)) < - (p `1) ) by XREAL_1:24;
assume  ( not ( p `1 <= p `2 & - (p `2) <= p `1 ) & not ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ; ::_thesis: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]|
hence  Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A2, A1, A14, Def1; ::_thesis: verum
end;

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 )
proof
let X be non empty TopSpace; ::_thesis: 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 )
let f1 be Function of X,R^1; ::_thesis: ( f1 is continuous & ( for q being Point of X ex r being real number st
( f1 . q = r & r >= 0 ) ) 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 = sqrt r1 ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: for q being Point of X ex r being real number st
( f1 . q = r & r >= 0 ) ; ::_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 = sqrt r1 ) & g is continuous )
defpred S1[ set , set ] means  for r11 being real number st f1 . $1 = r11 holds
$2 = sqrt r11;
A3: for x being Element of X ex y being Element of REAL st S1[x,y]
proof
let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider pp = x as Point of X ;
reconsider r1 = f1 . pp as Real by TOPMETR:17;
 for r11 being real number st f1 . x = r11 holds
sqrt r1 = sqrt r11 ;
hence  ex y being Element of REAL st S1[x,y] ; ::_thesis: verum
end;
 ex f being Function of the carrier of X,REAL st
for x2 being Element of X holds S1[x2,f . x2] from FUNCT_2:sch_3(A3);
then consider f being Function of the carrier of X,REAL such that
A4: for x2 being Element of X
for r11 being real number st f1 . x2 = r11 holds
f . x2 = sqrt r11 ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
 for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
reconsider r = g0 . p as Real by TOPMETR:17;
reconsider r1 = f1 . p as Real by TOPMETR:17;
assume  ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
then consider r01 being Real such that
A5: r01 > 0 and
A6: ].(r - r01),(r + r01).[ c= V by FRECHET:8;
set r0 = min (r01,1);
A7: min (r01,1) > 0 by A5, XXREAL_0:21;
A8: min (r01,1) > 0 by A5, XXREAL_0:21;
 min (r01,1) <= r01 by XXREAL_0:17;
then  ( r - r01 <= r - (min (r01,1)) & r + (min (r01,1)) <= r + r01 ) by XREAL_1:6, XREAL_1:10;
then  ].(r - (min (r01,1))),(r + (min (r01,1))).[ c= ].(r - r01),(r + r01).[ by XXREAL_1:46;
then A9: ].(r - (min (r01,1))),(r + (min (r01,1))).[ c= V by A6, XBOOLE_1:1;
A10: ex r8 being real number st
( f1 . p = r8 & r8 >= 0 ) by A2;
A11: r = sqrt r1 by A4;
then A12: r1 = r ^2 by A10, SQUARE_1:def_2;
A13: r >= 0 by A10, A11, SQUARE_1:17, SQUARE_1:26;
then A14: ((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2) > 0 + 0 by A8, SQUARE_1:12, XREAL_1:8;
percases ( r - (min (r01,1)) > 0 or r - (min (r01,1)) <= 0 ) ;
supposeA15: r - (min (r01,1)) > 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = (min (r01,1)) * (r - (min (r01,1)));
reconsider G1 = ].(r1 - ((min (r01,1)) * (r - (min (r01,1))))),(r1 + ((min (r01,1)) * (r - (min (r01,1))))).[ as Subset of R^1 by TOPMETR:17;
A16: r1 < r1 + ((min (r01,1)) * (r - (min (r01,1)))) by A8, A15, XREAL_1:29, XREAL_1:129;
then  r1 - ((min (r01,1)) * (r - (min (r01,1)))) < r1 by XREAL_1:19;
then A17: f1 . p in G1 by A16, XXREAL_1:4;
 G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A18: ( p in W1 & W1 is open ) and
A19: f1 .: W1 c= G1 by A1, A17, JGRAPH_2:10;
set W = W1;
A20: ( (r - ((1 / 2) * (min (r01,1)))) ^2 >= 0 & (min (r01,1)) ^2 >= 0 ) by XREAL_1:63;
now__::_thesis:_not_r1_=_0
assume  r1 = 0 ; ::_thesis: contradiction
then  r = 0 by A4, SQUARE_1:17;
hence  contradiction by A7, A15; ::_thesis: verum
end;
then  0 < r by A10, A11, SQUARE_1:25;
then A21: (min (r01,1)) * r > 0 by A8, XREAL_1:129;
then  0 + (r * (min (r01,1))) < (r * (min (r01,1))) + (r * (min (r01,1))) by XREAL_1:8;
then  ((min (r01,1)) * r) - ((min (r01,1)) * (min (r01,1))) < ((2 * r) * (min (r01,1))) - ((min (r01,1)) * (min (r01,1))) by XREAL_1:14;
then  - ((min (r01,1)) * (r - (min (r01,1)))) > - (((2 * r) * (min (r01,1))) - ((min (r01,1)) ^2)) by XREAL_1:24;
then  r1 + (- ((min (r01,1)) * (r - (min (r01,1))))) > (r ^2) + (- (((2 * r) * (min (r01,1))) - ((min (r01,1)) ^2))) by A12, XREAL_1:8;
then  sqrt (r1 - ((min (r01,1)) * (r - (min (r01,1))))) > sqrt ((r - (min (r01,1))) ^2) by SQUARE_1:27, XREAL_1:63;
then A22: sqrt (r1 - ((min (r01,1)) * (r - (min (r01,1))))) > r - (min (r01,1)) by A15, SQUARE_1:22;
 0 + (r * (min (r01,1))) < (r * (min (r01,1))) + (r * (min (r01,1))) by A21, XREAL_1:8;
then  ((min (r01,1)) * r) + 0 < ((2 * r) * (min (r01,1))) + (2 * ((min (r01,1)) * (min (r01,1)))) by A8, XREAL_1:8;
then  (((min (r01,1)) * r) - ((min (r01,1)) * (min (r01,1)))) + ((min (r01,1)) * (min (r01,1))) < (((2 * r) * (min (r01,1))) + ((min (r01,1)) * (min (r01,1)))) + ((min (r01,1)) * (min (r01,1))) ;
then  ((min (r01,1)) * r) - ((min (r01,1)) * (min (r01,1))) < ((2 * r) * (min (r01,1))) + ((min (r01,1)) * (min (r01,1))) by XREAL_1:7;
then  r1 + ((min (r01,1)) * (r - (min (r01,1)))) < (r ^2) + (((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) by A12, XREAL_1:8;
then  sqrt (r1 + ((min (r01,1)) * (r - (min (r01,1))))) < sqrt ((r + (min (r01,1))) ^2) by A10, A8, A15, SQUARE_1:27;
then A23: r + (min (r01,1)) > sqrt (r1 + ((min (r01,1)) * (r - (min (r01,1))))) by A13, A7, SQUARE_1:22;
A24: r1 - ((min (r01,1)) * (r - (min (r01,1)))) = (r ^2) - (((min (r01,1)) * r) - ((min (r01,1)) * (min (r01,1)))) by A10, A11, SQUARE_1:def_2
.= ((r - ((1 / 2) * (min (r01,1)))) ^2) + ((3 / 4) * ((min (r01,1)) ^2)) ;
 g0 .: W1 c= ].(r - (min (r01,1))),(r + (min (r01,1))).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - (min (r01,1))),(r + (min (r01,1))).[ )
assume  x in g0 .: W1 ; ::_thesis: x in ].(r - (min (r01,1))),(r + (min (r01,1))).[
then consider z being set  such that
A25: z in dom g0 and
A26: z in W1 and
A27: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A25;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
A28: ex r9 being real number st
( f1 . pz = r9 & r9 >= 0 ) by A2;
 pz in the carrier of X ;
then  pz in dom f1 by FUNCT_2:def_1;
then A29: f1 . pz in f1 .: W1 by A26, FUNCT_1:def_6;
then  aa1 < r1 + ((min (r01,1)) * (r - (min (r01,1)))) by A19, XXREAL_1:4;
then  sqrt aa1 < sqrt (r1 + ((min (r01,1)) * (r - (min (r01,1))))) by A28, SQUARE_1:27;
then A30: sqrt aa1 < r + (min (r01,1)) by A23, XXREAL_0:2;
A31: r1 - ((min (r01,1)) * (r - (min (r01,1)))) < aa1 by A19, A29, XXREAL_1:4;
A32: now__::_thesis:_(_(_0_<=_r1_-_((min_(r01,1))_*_(r_-_(min_(r01,1))))_&_r_-_(min_(r01,1))_<_sqrt_aa1_)_or_(_0_>_r1_-_((min_(r01,1))_*_(r_-_(min_(r01,1))))_&_contradiction_)_)
percases ( 0 <= r1 - ((min (r01,1)) * (r - (min (r01,1)))) or 0 > r1 - ((min (r01,1)) * (r - (min (r01,1)))) ) ;
case 0 <= r1 - ((min (r01,1)) * (r - (min (r01,1)))) ; ::_thesis: r - (min (r01,1)) < sqrt aa1
then  sqrt (r1 - ((min (r01,1)) * (r - (min (r01,1))))) <= sqrt aa1 by A31, SQUARE_1:26;
hence  r - (min (r01,1)) < sqrt aa1 by A22, XXREAL_0:2; ::_thesis: verum
end;
case 0 > r1 - ((min (r01,1)) * (r - (min (r01,1)))) ; ::_thesis: contradiction
hence  contradiction by A24, A20; ::_thesis: verum
end;
end;
end;
 x = sqrt aa1 by A4, A27;
hence  x in ].(r - (min (r01,1))),(r + (min (r01,1))).[ by A30, A32, XXREAL_1:4; ::_thesis: verum
end;
hence  ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A9, A18, XBOOLE_1:1; ::_thesis: verum
end;
supposeA33: r - (min (r01,1)) <= 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = (((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3;
reconsider G1 = ].(r1 - ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3)),(r1 + ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3)).[ as Subset of R^1 by TOPMETR:17;
 (((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3 > 0 by A14, XREAL_1:139;
then A34: r1 < r1 + ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3) by XREAL_1:29;
then  r1 - ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3) < r1 by XREAL_1:19;
then A35: f1 . p in G1 by A34, XXREAL_1:4;
 G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A36: ( p in W1 & W1 is open ) and
A37: f1 .: W1 c= G1 by A1, A35, JGRAPH_2:10;
set W = W1;
 (((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3 < ((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2) by A14, XREAL_1:221;
then  r1 + ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3) < (r ^2) + (((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) by A12, XREAL_1:8;
then  sqrt (r1 + ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3)) <= sqrt ((r + (min (r01,1))) ^2) by A10, A13, A8, SQUARE_1:26;
then A38: r + (min (r01,1)) >= sqrt (r1 + ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3)) by A13, A7, SQUARE_1:22;
 g0 .: W1 c= ].(r - (min (r01,1))),(r + (min (r01,1))).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - (min (r01,1))),(r + (min (r01,1))).[ )
assume  x in g0 .: W1 ; ::_thesis: x in ].(r - (min (r01,1))),(r + (min (r01,1))).[
then consider z being set  such that
A39: z in dom g0 and
A40: z in W1 and
A41: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A39;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
A42: ex r9 being real number st
( f1 . pz = r9 & r9 >= 0 ) by A2;
 pz in the carrier of X ;
then  pz in dom f1 by FUNCT_2:def_1;
then A43: f1 . pz in f1 .: W1 by A40, FUNCT_1:def_6;
then  aa1 < r1 + ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3) by A37, XXREAL_1:4;
then  sqrt aa1 < sqrt (r1 + ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3)) by A42, SQUARE_1:27;
then A44: sqrt aa1 < r + (min (r01,1)) by A38, XXREAL_0:2;
A45: r1 - ((((2 * r) * (min (r01,1))) + ((min (r01,1)) ^2)) / 3) < aa1 by A37, A43, XXREAL_1:4;
A46: now__::_thesis:_(_(_r_-_(min_(r01,1))_=_0_&_r_-_(min_(r01,1))_<_sqrt_aa1_)_or_(_r_-_(min_(r01,1))_<_0_&_r_-_(min_(r01,1))_<_sqrt_aa1_)_)
percases ( r - (min (r01,1)) = 0 or r - (min (r01,1)) < 0 ) by A33;
case r - (min (r01,1)) = 0 ; ::_thesis: r - (min (r01,1)) < sqrt aa1
hence  r - (min (r01,1)) < sqrt aa1 by A12, A45, SQUARE_1:17, SQUARE_1:27; ::_thesis: verum
end;
case r - (min (r01,1)) < 0 ; ::_thesis: r - (min (r01,1)) < sqrt aa1
hence  r - (min (r01,1)) < sqrt aa1 by A42, SQUARE_1:17, SQUARE_1:26; ::_thesis: verum
end;
end;
end;
 x = sqrt aa1 by A4, A41;
hence  x in ].(r - (min (r01,1))),(r + (min (r01,1))).[ by A44, A46, XXREAL_1:4; ::_thesis: verum
end;
hence  ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A9, A36, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
then A47: g0 is continuous by JGRAPH_2:10;
 for p being Point of X
for r11 being real number st f1 . p = r11 holds
g0 . p = sqrt r11 by A4;
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 = sqrt r1 ) & g is continuous ) by A47; ::_thesis: verum
end;

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 )
proof
let X be non empty TopSpace; ::_thesis: 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 )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( 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) ^2 ) & g is continuous ) )
assume  ( f1 is continuous & f2 is continuous & ( 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) ^2 ) & g is continuous )
then consider g2 being Function of X,R^1 such that
A1: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = r1 / r2 and
A2: g2 is continuous by JGRAPH_2:27;
consider g3 being Function of X,R^1 such that
A3: for p being Point of X
for r1 being real number st g2 . p = r1 holds
g3 . p = r1 * r1 and
A4: g3 is continuous by A2, JGRAPH_2:22;
 for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = (r1 / r2) ^2
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = (r1 / r2) ^2
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = (r1 / r2) ^2 )
assume  ( f1 . p = r1 & f2 . p = r2 ) ; ::_thesis: g3 . p = (r1 / r2) ^2
then  g2 . p = r1 / r2 by A1;
hence  g3 . p = (r1 / r2) ^2 by A3; ::_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) ^2 ) & g is continuous ) by A4; ::_thesis: verum
end;

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 )
proof
let X be non empty TopSpace; ::_thesis: 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 )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( 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 = 1 + ((r1 / r2) ^2) ) & g is continuous ) )
assume  ( f1 is continuous & f2 is continuous & ( 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 = 1 + ((r1 / r2) ^2) ) & g is continuous )
then consider g2 being Function of X,R^1 such that
A1: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = (r1 / r2) ^2 and
A2: g2 is continuous by Th6;
consider g3 being Function of X,R^1 such that
A3: for p being Point of X
for r1 being real number st g2 . p = r1 holds
g3 . p = r1 + 1 and
A4: g3 is continuous by A2, JGRAPH_2:24;
 for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = 1 + ((r1 / r2) ^2)
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = 1 + ((r1 / r2) ^2)
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = 1 + ((r1 / r2) ^2) )
assume  ( f1 . p = r1 & f2 . p = r2 ) ; ::_thesis: g3 . p = 1 + ((r1 / r2) ^2)
then  g2 . p = (r1 / r2) ^2 by A1;
hence  g3 . p = 1 + ((r1 / r2) ^2) by A3; ::_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 = 1 + ((r1 / r2) ^2) ) & g is continuous ) by A4; ::_thesis: verum
end;

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 )
proof
let X be non empty TopSpace; ::_thesis: 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 )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( 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 = sqrt (1 + ((r1 / r2) ^2)) ) & g is continuous ) )
assume  ( f1 is continuous & f2 is continuous & ( 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 = sqrt (1 + ((r1 / r2) ^2)) ) & g is continuous )
then consider g2 being Function of X,R^1 such that
A1: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = 1 + ((r1 / r2) ^2) and
A2: g2 is continuous by Th7;
 for q being Point of X ex r being real number st
( g2 . q = r & r >= 0 )
proof
let q be Point of X; ::_thesis: ex r being real number st
( g2 . q = r & r >= 0 )
reconsider r1 = f1 . q, r2 = f2 . q as Real by TOPMETR:17;
 1 + ((r1 / r2) ^2) > 0 by Lm1;
hence  ex r being real number st
( g2 . q = r & r >= 0 ) by A1; ::_thesis: verum
end;
then consider g3 being Function of X,R^1 such that
A3: for p being Point of X
for r1 being real number st g2 . p = r1 holds
g3 . p = sqrt r1 and
A4: g3 is continuous by A2, Th5;
 for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = sqrt (1 + ((r1 / r2) ^2))
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = sqrt (1 + ((r1 / r2) ^2))
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = sqrt (1 + ((r1 / r2) ^2)) )
assume  ( f1 . p = r1 & f2 . p = r2 ) ; ::_thesis: g3 . p = sqrt (1 + ((r1 / r2) ^2))
then  g2 . p = 1 + ((r1 / r2) ^2) by A1;
hence  g3 . p = sqrt (1 + ((r1 / r2) ^2)) by A3; ::_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 = sqrt (1 + ((r1 / r2) ^2)) ) & g is continuous ) by A4; ::_thesis: verum
end;

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 )
proof
let X be non empty TopSpace; ::_thesis: 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 )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( 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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: ( f2 is continuous & ( 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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous )
consider g2 being Function of X,R^1 such that
A3: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = sqrt (1 + ((r1 / r2) ^2)) and
A4: g2 is continuous by A1, A2, Th8;
 for q being Point of X holds g2 . q <> 0
proof
let q be Point of X; ::_thesis: g2 . q <> 0
reconsider r1 = f1 . q, r2 = f2 . q as Real by TOPMETR:17;
 sqrt (1 + ((r1 / r2) ^2)) > 0 by Lm1, SQUARE_1:25;
hence  g2 . q <> 0 by A3; ::_thesis: verum
end;
then 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 & g2 . p = r0 holds
g3 . p = r1 / r0 and
A6: g3 is continuous by A1, A4, JGRAPH_2:27;
 for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2)))
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2)))
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2))) )
assume that
A7: f1 . p = r1 and
A8: f2 . p = r2 ; ::_thesis: g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2)))
 g2 . p = sqrt (1 + ((r1 / r2) ^2)) by A3, A7, A8;
hence  g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2))) by A5, A7; ::_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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) by A6; ::_thesis: verum
end;

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 )
proof
let X be non empty TopSpace; ::_thesis: 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 )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( 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 (1 + ((r1 / r2) ^2))) ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: f2 is continuous 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 = r2 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous )
consider g2 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
g2 . p = sqrt (1 + ((r1 / r2) ^2)) and
A5: g2 is continuous by A1, A2, A3, Th8;
 for q being Point of X holds g2 . q <> 0
proof
let q be Point of X; ::_thesis: g2 . q <> 0
reconsider r1 = f1 . q, r2 = f2 . q as Real by TOPMETR:17;
 sqrt (1 + ((r1 / r2) ^2)) > 0 by Lm1, SQUARE_1:25;
hence  g2 . q <> 0 by A4; ::_thesis: verum
end;
then consider g3 being Function of X,R^1 such that
A6: for p being Point of X
for r2, r0 being real number st f2 . p = r2 & g2 . p = r0 holds
g3 . p = r2 / r0 and
A7: g3 is continuous by A2, A5, JGRAPH_2:27;
 for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = r2 / (sqrt (1 + ((r1 / r2) ^2)))
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = r2 / (sqrt (1 + ((r1 / r2) ^2)))
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = r2 / (sqrt (1 + ((r1 / r2) ^2))) )
assume that
A8: f1 . p = r1 and
A9: f2 . p = r2 ; ::_thesis: g3 . p = r2 / (sqrt (1 + ((r1 / r2) ^2)))
 g2 . p = sqrt (1 + ((r1 / r2) ^2)) by A4, A8, A9;
hence  g3 . p = r2 / (sqrt (1 + ((r1 / r2) ^2))) by A6, A9; ::_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 (1 + ((r1 / r2) ^2))) ) & g is continuous ) by A7; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for q being Point of ((TOP-REAL 2) | K1) holds (proj2 | K1) . q = proj2 . q
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: (proj2 | K1) . q = proj2 . q
 ( the carrier of ((TOP-REAL 2) | K1) = K1 & q in the carrier of ((TOP-REAL 2) | K1) ) by PRE_TOPC:8;
then  q in (dom proj2) /\ K1 by Lm3, XBOOLE_0:def_4;
hence  (proj2 | K1) . q = proj2 . q by FUNCT_1:48; ::_thesis: verum
end;

Lm5:  for K1 being non empty Subset of (TOP-REAL 2) holds proj2 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: proj2 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1
reconsider g2 = proj2 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
 for q being Point of ((TOP-REAL 2) | K1) holds g2 . q = proj2 . q by Lm4;
hence  proj2 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1 by JGRAPH_2:30; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for q being Point of ((TOP-REAL 2) | K1) holds (proj1 | K1) . q = proj1 . q
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: (proj1 | K1) . q = proj1 . q
 ( the carrier of ((TOP-REAL 2) | K1) = K1 & q in the carrier of ((TOP-REAL 2) | K1) ) by PRE_TOPC:8;
then  q in (dom proj1) /\ K1 by Lm2, XBOOLE_0:def_4;
hence  (proj1 | K1) . q = proj1 . q by FUNCT_1:48; ::_thesis: verum
end;

Lm7:  for K1 being non empty Subset of (TOP-REAL 2) holds proj1 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: proj1 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1
reconsider g2 = proj1 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
 for q being Point of ((TOP-REAL 2) | K1) holds g2 . q = proj1 . q by Lm6;
hence  proj1 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1 by JGRAPH_2:29; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( 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 ) implies f is continuous )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
A1: 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))) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ; ::_thesis: f is continuous
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
now__::_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
 q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj1 . q by Lm6
.= q2 `1 by PSCOMP_1:def_5 ;
hence  g1 . q <> 0 by A2; ::_thesis: verum
end;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: 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 (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th10;
A6: 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 A7: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
 x in the carrier of ((TOP-REAL 2) | K1) by A7;
then  x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
 f . r = (r `1) / (sqrt (1 + (((r `2) / (r `1)) ^2))) by A1, A7;
hence  f . x = g3 . x by A4, A9, A8; ::_thesis: verum
end;
 dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then  dom f = dom g3 by FUNCT_2:def_1;
hence  f is continuous by A5, A6, FUNCT_1:2; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( 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 ) implies f is continuous )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
A1: 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))) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ; ::_thesis: f is continuous
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
now__::_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
 q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj1 . q by Lm6
.= q2 `1 by PSCOMP_1:def_5 ;
hence  g1 . q <> 0 by A2; ::_thesis: verum
end;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: 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 = r1 / (sqrt (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th9;
A6: 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 A7: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
 x in the carrier of ((TOP-REAL 2) | K1) by A7;
then  x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
 f . r = (r `2) / (sqrt (1 + (((r `2) / (r `1)) ^2))) by A1, A7;
hence  f . x = g3 . x by A4, A9, A8; ::_thesis: verum
end;
 dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then  dom f = dom g3 by FUNCT_2:def_1;
hence  f is continuous by A5, A6, FUNCT_1:2; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( 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 ) implies f is continuous )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
A1: 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))) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ; ::_thesis: f is continuous
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
now__::_thesis:_for_q_being_Point_of_((TOP-REAL_2)_|_K1)_holds_g2_._q_<>_0
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g2 . q <> 0
 q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g2 . q = proj2 . q by Lm4
.= q2 `2 by PSCOMP_1:def_6 ;
hence  g2 . q <> 0 by A2; ::_thesis: verum
end;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being real number st g1 . q = r1 & g2 . q = r2 holds
g3 . q = r2 / (sqrt (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th10;
A6: 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 A7: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
 x in the carrier of ((TOP-REAL 2) | K1) by A7;
then  x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
 f . r = (r `2) / (sqrt (1 + (((r `1) / (r `2)) ^2))) by A1, A7;
hence  f . x = g3 . x by A4, A9, A8; ::_thesis: verum
end;
 dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then  dom f = dom g3 by FUNCT_2:def_1;
hence  f is continuous by A5, A6, FUNCT_1:2; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( 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 ) implies f is continuous )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
A1: 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))) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ; ::_thesis: f is continuous
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
now__::_thesis:_for_q_being_Point_of_((TOP-REAL_2)_|_K1)_holds_g2_._q_<>_0
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g2 . q <> 0
 q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g2 . q = proj2 . q by Lm4
.= q2 `2 by PSCOMP_1:def_6 ;
hence  g2 . q <> 0 by A2; ::_thesis: verum
end;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being real number st g1 . q = r1 & g2 . q = r2 holds
g3 . q = r1 / (sqrt (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th9;
A6: 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 A7: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
 x in the carrier of ((TOP-REAL 2) | K1) by A7;
then  x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
 f . r = (r `1) / (sqrt (1 + (((r `1) / (r `2)) ^2))) by A1, A7;
hence  f . x = g3 . x by A4, A9, A8; ::_thesis: verum
end;
 dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then  dom f = dom g3 by FUNCT_2:def_1;
hence  f is continuous by A5, A6, FUNCT_1:2; ::_thesis: verum
end;

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)
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: dom (proj2 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1)
A1: dom (Sq_Circ | K1) c= dom (proj2 * (Sq_Circ | K1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Sq_Circ | K1) or x in dom (proj2 * (Sq_Circ | K1)) )
assume A2: x in dom (Sq_Circ | K1) ; ::_thesis: x in dom (proj2 * (Sq_Circ | K1))
then  x in (dom Sq_Circ) /\ K1 by RELAT_1:61;
then  x in dom Sq_Circ by XBOOLE_0:def_4;
then A3: Sq_Circ . x in rng Sq_Circ by FUNCT_1:3;
 (Sq_Circ | K1) . x = Sq_Circ . x by A2, FUNCT_1:47;
hence  x in dom (proj2 * (Sq_Circ | K1)) by A2, A3, Lm3, FUNCT_1:11; ::_thesis: verum
end;
 dom (proj2 * (Sq_Circ | K1)) c= dom (Sq_Circ | K1) by RELAT_1:25;
hence  dom (proj2 * (Sq_Circ | K1)) = dom (Sq_Circ | K1) by A1, XBOOLE_0:def_10
.= (dom Sq_Circ) /\ 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 ;
::_thesis: verum
end;

Lm12:  for K1 being non empty Subset of (TOP-REAL 2) holds dom (proj1 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1)
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: dom (proj1 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1)
A1: dom (Sq_Circ | K1) c= dom (proj1 * (Sq_Circ | K1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Sq_Circ | K1) or x in dom (proj1 * (Sq_Circ | K1)) )
assume A2: x in dom (Sq_Circ | K1) ; ::_thesis: x in dom (proj1 * (Sq_Circ | K1))
then  x in (dom Sq_Circ) /\ K1 by RELAT_1:61;
then  x in dom Sq_Circ by XBOOLE_0:def_4;
then A3: Sq_Circ . x in rng Sq_Circ by FUNCT_1:3;
 (Sq_Circ | K1) . x = Sq_Circ . x by A2, FUNCT_1:47;
hence  x in dom (proj1 * (Sq_Circ | K1)) by A2, A3, Lm2, FUNCT_1:11; ::_thesis: verum
end;
 dom (proj1 * (Sq_Circ | K1)) c= dom (Sq_Circ | K1) by RELAT_1:25;
hence  dom (proj1 * (Sq_Circ | K1)) = dom (Sq_Circ | K1) by A1, XBOOLE_0:def_10
.= (dom Sq_Circ) /\ 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 ;
::_thesis: verum
end;

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
proof
let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( 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) )  }  implies f is continuous )
assume A1: ( 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) )  }  ) ; ::_thesis: f is continuous
then  1.REAL 2 in K0 by Lm9, Lm10;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
 ( dom (proj1 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1) & rng (proj1 * (Sq_Circ | K1)) c= the carrier of R^1 ) by Lm12, TOPMETR:17;
then reconsider g1 = proj1 * (Sq_Circ | K1) as Function of ((TOP-REAL 2) | K1),R^1 by 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 `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) )
A2: dom (Sq_Circ | K1) = (dom Sq_Circ) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1
.= K1 by XBOOLE_1:28 ;
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
assume A4: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))
then  ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A3;
then A5: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by Def1;
 (Sq_Circ | K1) . p = Sq_Circ . p by A4, A3, FUNCT_1:49;
then g1 . p = proj1 . |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A4, A2, A3, A5, FUNCT_1:13
.= |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 by PSCOMP_1:def_5
.= (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52 ;
hence  g1 . p = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ; ::_thesis: verum
end;
then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that
A6: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f1 . p = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ;
 ( dom (proj2 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1) & rng (proj2 * (Sq_Circ | K1)) c= the carrier of R^1 ) by Lm11, TOPMETR:17;
then reconsider g2 = proj2 * (Sq_Circ | K1) as Function of ((TOP-REAL 2) | K1),R^1 by 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 `2) / (sqrt (1 + (((p `2) / (p `1)) ^2)))
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) )
A7: dom (Sq_Circ | K1) = (dom Sq_Circ) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1
.= K1 by XBOOLE_1:28 ;
A8: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
assume A9: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2)))
then  ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A8;
then A10: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by Def1;
 (Sq_Circ | K1) . p = Sq_Circ . p by A9, A8, FUNCT_1:49;
then g2 . p = proj2 . |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A9, A7, A8, A10, FUNCT_1:13
.= |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 by PSCOMP_1:def_6
.= (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52 ;
hence  g2 . p = (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ; ::_thesis: verum
end;
then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that
A11: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f2 . p = (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ;
A12: now__::_thesis:_for_q_being_Point_of_(TOP-REAL_2)_st_q_in_the_carrier_of_((TOP-REAL_2)_|_K1)_holds_
q_`1_<>_0
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies q `1 <> 0 )
A13: 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
then A14: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A13;
now__::_thesis:_not_q_`1_=_0
assume A15: q `1 = 0 ; ::_thesis: contradiction
then  q `2 = 0 by A14;
hence  contradiction by A14, A15, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence  q `1 <> 0 ; ::_thesis: verum
end;
then A16: f1 is continuous by A6, Th11;
A17: 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
A18: |[x,y]| in K1 and
A19: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[r,s]|
set p99 = |[x,y]|;
A20: ex p3 being Point of (TOP-REAL 2) st
( |[x,y]| = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A18;
A21: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
then A22: f1 . |[x,y]| = (|[x,y]| `1) / (sqrt (1 + (((|[x,y]| `2) / (|[x,y]| `1)) ^2))) by A6, A18;
(Sq_Circ | K0) . |[x,y]| = Sq_Circ . |[x,y]| by A18, FUNCT_1:49
.= |[((|[x,y]| `1) / (sqrt (1 + (((|[x,y]| `2) / (|[x,y]| `1)) ^2)))),((|[x,y]| `2) / (sqrt (1 + (((|[x,y]| `2) / (|[x,y]| `1)) ^2))))]| by A20, Def1
.= |[r,s]| by A11, A18, A19, A21, A22 ;
hence  f . |[x,y]| = |[r,s]| by A1; ::_thesis: verum
end;
 f2 is continuous by A12, A11, Th12;
hence  f is continuous by A1, A16, A17, Lm13, JGRAPH_2:35; ::_thesis: verum
end;

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
proof
let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( 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) )  }  implies f is continuous )
assume A1: ( 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) )  }  ) ; ::_thesis: f is continuous
then  1.REAL 2 in K0 by Lm14, Lm15;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
 ( dom (proj2 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1) & rng (proj2 * (Sq_Circ | K1)) c= the carrier of R^1 ) by Lm11, TOPMETR:17;
then reconsider g1 = proj2 * (Sq_Circ | K1) as Function of ((TOP-REAL 2) | K1),R^1 by 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 `2) / (sqrt (1 + (((p `1) / (p `2)) ^2)))
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) )
A2: dom (Sq_Circ | K1) = (dom Sq_Circ) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1
.= K1 by XBOOLE_1:28 ;
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
assume A4: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2)))
then  ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A3;
then A5: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by Th4;
 (Sq_Circ | K1) . p = Sq_Circ . p by A4, A3, FUNCT_1:49;
then g1 . p = proj2 . |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A4, A2, A3, A5, FUNCT_1:13
.= |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 by PSCOMP_1:def_6
.= (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52 ;
hence  g1 . p = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ; ::_thesis: verum
end;
then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that
A6: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f1 . p = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ;
 ( dom (proj1 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1) & rng (proj1 * (Sq_Circ | K1)) c= the carrier of R^1 ) by Lm12, TOPMETR:17;
then reconsider g2 = proj1 * (Sq_Circ | K1) as Function of ((TOP-REAL 2) | K1),R^1 by 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 `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) )
A7: dom (Sq_Circ | K1) = (dom Sq_Circ) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1
.= K1 by XBOOLE_1:28 ;
A8: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
assume A9: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))
then  ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A8;
then A10: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by Th4;
 (Sq_Circ | K1) . p = Sq_Circ . p by A9, A8, FUNCT_1:49;
then g2 . p = proj1 . |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A9, A7, A8, A10, FUNCT_1:13
.= |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 by PSCOMP_1:def_5
.= (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52 ;
hence  g2 . p = (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ; ::_thesis: verum
end;
then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that
A11: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f2 . p = (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ;
A12: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies q `2 <> 0 )
A13: 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
then A14: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A13;
now__::_thesis:_not_q_`2_=_0
assume A15: q `2 = 0 ; ::_thesis: contradiction
then  q `1 = 0 by A14;
hence  contradiction by A14, A15, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence  q `2 <> 0 ; ::_thesis: verum
end;
then A16: f1 is continuous by A6, Th13;
A17: now__::_thesis:_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]|
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
A18: |[x,y]| in K1 and
A19: ( s = f2 . |[x,y]| & r = f1 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[s,r]|
set p99 = |[x,y]|;
A20: ex p3 being Point of (TOP-REAL 2) st
( |[x,y]| = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A18;
A21: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
then A22: f1 . |[x,y]| = (|[x,y]| `2) / (sqrt (1 + (((|[x,y]| `1) / (|[x,y]| `2)) ^2))) by A6, A18;
(Sq_Circ | K0) . |[x,y]| = Sq_Circ . |[x,y]| by A18, FUNCT_1:49
.= |[((|[x,y]| `1) / (sqrt (1 + (((|[x,y]| `1) / (|[x,y]| `2)) ^2)))),((|[x,y]| `2) / (sqrt (1 + (((|[x,y]| `1) / (|[x,y]| `2)) ^2))))]| by A20, Th4
.= |[s,r]| by A11, A18, A19, A21, A22 ;
hence  f . |[x,y]| = |[s,r]| by A1; ::_thesis: verum
end;
 f2 is continuous by A12, A11, Th14;
hence  f is continuous by A1, A16, A17, Lm13, JGRAPH_2:35; ::_thesis: verum
end;

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)
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 NonZero (TOP-REAL 2) )
assume  x in  {  p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) )  }  ; ::_thesis: x in NonZero (TOP-REAL 2)
then A1: ex p8 being Point of (TOP-REAL 2) st
( x = p8 & P1[p8] & p8 <> 0. (TOP-REAL 2) ) ;
then  not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence  x in NonZero (TOP-REAL 2) by A1, XBOOLE_0:def_5; ::_thesis: verum
end;

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))
proof
set B0 = NonZero (TOP-REAL 2);
set K1 =  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  ;
set K0 =  {  p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) )  }  ;
A1:  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  /\ (NonZero (TOP-REAL 2)) c=  {  p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  /\ (NonZero (TOP-REAL 2)) or x in  {  p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) )  }  )
assume A2: x in  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  /\ (NonZero (TOP-REAL 2)) ; ::_thesis: x in  {  p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) )  }
then  x in NonZero (TOP-REAL 2) by XBOOLE_0:def_4;
then  not x in {(0. (TOP-REAL 2))} by XBOOLE_0:def_5;
then A3: x <> 0. (TOP-REAL 2) by TARSKI:def_1;
 x in  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  by A2, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & P1[p7] ) ;
hence  x in  {  p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) )  }  by A3; ::_thesis: verum
end;
  {  p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) )  }  c=  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  /\ (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  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  /\ (NonZero (TOP-REAL 2)) )
assume  x in  {  p where p is Point of (TOP-REAL 2) : ( P1[p] & p <> 0. (TOP-REAL 2) )  }  ; ::_thesis: x in  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  /\ (NonZero (TOP-REAL 2))
then A4: ex p being Point of (TOP-REAL 2) st
( x = p & P1[p] & p <> 0. (TOP-REAL 2) ) ;
then  not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
then A5: x in NonZero (TOP-REAL 2) by A4, XBOOLE_0:def_5;
 x in  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  by A4;
hence  x in  {  p7 where p7 is Point of (TOP-REAL 2) : P1[p7]  }  /\ (NonZero (TOP-REAL 2)) by A5, XBOOLE_0:def_4; ::_thesis: verum
end;
hence   {  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)) by A1, XBOOLE_0:def_10; ::_thesis: verum
end;

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 )
proof
reconsider K5 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1)  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:47;
reconsider K4 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
reconsider K3 =  {  p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1) <= p7 `2  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:47;
reconsider K2 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ) );
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 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 )
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 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 )
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( 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) )  }  implies ( f is continuous & K0 is closed ) )
assume A1: ( 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) )  }  ) ; ::_thesis: ( f is continuous & K0 is closed )
 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;
  {  p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) )  }  c= NonZero (TOP-REAL 2) from JGRAPH_3:sch_1();
then A2: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by A1, PRE_TOPC:7;
defpred S2[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `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();
defpred S3[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ) );
  {  p where p is Point of (TOP-REAL 2) : ( S3[p] & p <> 0. (TOP-REAL 2) )  }  =  {  p7 where p7 is Point of (TOP-REAL 2) : S3[p7]  }  /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch_2();
then A3: K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) by A1, PRE_TOPC:def_5;
A4: (K2 /\ K3) \/ (K4 /\ K5) c= K1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A5: x in (K2 /\ K3) \/ (K4 /\ K5) ; ::_thesis: x in K1
percases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A5, XBOOLE_0:def_3;
supposeA6: x in K2 /\ K3 ; ::_thesis: x in K1
then  x in K3 by XBOOLE_0:def_4;
then A7: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & - (p8 `1) <= p8 `2 ) ;
 x in K2 by A6, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `2 <= p7 `1 ) ;
hence  x in K1 by A7; ::_thesis: verum
end;
supposeA8: x in K4 /\ K5 ; ::_thesis: x in K1
then  x in K5 by XBOOLE_0:def_4;
then A9: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & p8 `2 <= - (p8 `1) ) ;
 x in K4 by A8, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `2 >= p7 `1 ) ;
hence  x in K1 by A9; ::_thesis: verum
end;
end;
end;
A10: ( K2 /\ K3 is closed & K4 /\ K5 is closed ) by TOPS_1:8;
 K1 c= (K2 /\ K3) \/ (K4 /\ K5)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) )
assume  x in K1 ; ::_thesis: x in (K2 /\ K3) \/ (K4 /\ K5)
then  ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ) ;
then  ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ;
then  ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def_4;
hence  x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def_3; ::_thesis: verum
end;
then  K1 = (K2 /\ K3) \/ (K4 /\ K5) by A4, XBOOLE_0:def_10;
then  K1 is closed by A10, TOPS_1:9;
hence  ( f is continuous & K0 is closed ) by A1, A2, A3, Th15, PRE_TOPC:13; ::_thesis: verum
end;

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 )
proof
reconsider K5 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= - (p7 `2)  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:48;
reconsider K4 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
reconsider K3 =  {  p7 where p7 is Point of (TOP-REAL 2) : - (p7 `2) <= p7 `1  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:48;
reconsider K2 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `1 <= $1 `2 & - ($1 `2) <= $1 `1 ) or ( $1 `1 >= $1 `2 & $1 `1 <= - ($1 `2) ) );
set b0 = NonZero (TOP-REAL 2);
defpred S2[ Point of (TOP-REAL 2)] means ( ( $1 `1 <= $1 `2 & - ($1 `2) <= $1 `1 ) or ( $1 `1 >= $1 `2 & $1 `1 <= - ($1 `2) ) );
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 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 )
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 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 )
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( 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) )  }  implies ( f is continuous & K0 is closed ) )
assume A1: ( 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) )  }  ) ; ::_thesis: ( f is continuous & K0 is closed )
 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;
  {  p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) )  }  c= NonZero (TOP-REAL 2) from JGRAPH_3:sch_1();
then A2: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by A1, PRE_TOPC:7;
defpred S3[ Point of (TOP-REAL 2)] means ( ( $1 `1 <= $1 `2 & - ($1 `2) <= $1 `1 ) or ( $1 `1 >= $1 `2 & $1 `1 <= - ($1 `2) ) );
reconsider K1 =  {  p7 where p7 is Point of (TOP-REAL 2) : S3[p7]  }  as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
A3: (K2 /\ K3) \/ (K4 /\ K5) c= K1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A4: x in (K2 /\ K3) \/ (K4 /\ K5) ; ::_thesis: x in K1
percases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A4, XBOOLE_0:def_3;
supposeA5: x in K2 /\ K3 ; ::_thesis: x in K1
then  x in K3 by XBOOLE_0:def_4;
then A6: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & - (p8 `2) <= p8 `1 ) ;
 x in K2 by A5, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `1 <= p7 `2 ) ;
hence  x in K1 by A6; ::_thesis: verum
end;
supposeA7: x in K4 /\ K5 ; ::_thesis: x in K1
then  x in K5 by XBOOLE_0:def_4;
then A8: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & p8 `1 <= - (p8 `2) ) ;
 x in K4 by A7, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `1 >= p7 `2 ) ;
hence  x in K1 by A8; ::_thesis: verum
end;
end;
end;
set k0 =  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  ;
A9: ( K2 /\ K3 is closed & K4 /\ K5 is closed ) by TOPS_1:8;
 K1 c= (K2 /\ K3) \/ (K4 /\ K5)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) )
assume  x in K1 ; ::_thesis: x in (K2 /\ K3) \/ (K4 /\ K5)
then  ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ) ;
then  ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ;
then  ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def_4;
hence  x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def_3; ::_thesis: verum
end;
then  K1 = (K2 /\ K3) \/ (K4 /\ K5) by A3, XBOOLE_0:def_10;
then A10: K1 is closed by A9, TOPS_1:9;
  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  =  {  p7 where p7 is Point of (TOP-REAL 2) : S2[p7]  }  /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch_2();
then  K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) by A1, PRE_TOPC:def_5;
hence  ( f is continuous & K0 is closed ) by A1, A2, A10, Th16, PRE_TOPC:13; ::_thesis: verum
end;

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 )
proof
set Y1 = |[(- 1),1]|;
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( D ` = {(0. (TOP-REAL 2))} implies ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Sq_Circ | D & h is continuous ) )
A1: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8;
 dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then A2: dom (Sq_Circ | D) = the carrier of (TOP-REAL 2) /\ D by RELAT_1:61
.= the carrier of ((TOP-REAL 2) | D) by A1, XBOOLE_1:28 ;
assume A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Sq_Circ | D & h is continuous )
then A4: D = {(0. (TOP-REAL 2))} `
.= NonZero (TOP-REAL 2) by SUBSET_1:def_4 ;
A5:  {  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) )  }  c= the carrier of ((TOP-REAL 2) | D)
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 `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) )  }  or x in the carrier of ((TOP-REAL 2) | D) )
assume  x in  {  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) )  }  ; ::_thesis: x in the carrier of ((TOP-REAL 2) | D)
then A6: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) ;
now__::_thesis:_x_in_D
assume  not x in D ; ::_thesis: contradiction
then  x in the carrier of (TOP-REAL 2) \ D by A6, XBOOLE_0:def_5;
then  x in D ` by SUBSET_1:def_4;
hence  contradiction by A3, A6, TARSKI:def_1; ::_thesis: verum
end;
hence  x in the carrier of ((TOP-REAL 2) | D) by PRE_TOPC:8; ::_thesis: verum
end;
 1.REAL 2 in  {  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) )  }  by Lm9, Lm10;
then reconsider 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) )  }  as non empty Subset of ((TOP-REAL 2) | D) by A5;
A7: K0 = the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8;
A8:  {  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) )  }  c= the carrier of ((TOP-REAL 2) | D)
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 `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) )  }  or x in the carrier of ((TOP-REAL 2) | D) )
assume  x in  {  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) )  }  ; ::_thesis: x in the carrier of ((TOP-REAL 2) | D)
then A9: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) ;
now__::_thesis:_x_in_D
assume  not x in D ; ::_thesis: contradiction
then  x in the carrier of (TOP-REAL 2) \ D by A9, XBOOLE_0:def_5;
then  x in D ` by SUBSET_1:def_4;
hence  contradiction by A3, A9, TARSKI:def_1; ::_thesis: verum
end;
hence  x in the carrier of ((TOP-REAL 2) | D) by PRE_TOPC:8; ::_thesis: verum
end;
 ( |[(- 1),1]| `1 = - 1 & |[(- 1),1]| `2 = 1 ) by EUCLID:52;
then  |[(- 1),1]| in  {  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) )  }  by JGRAPH_2:3;
then reconsider K1 =  {  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) )  }  as non empty Subset of ((TOP-REAL 2) | D) by A8;
A10: K1 = the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8;
A11: 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 A12: 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 A4, A12, XBOOLE_0:def_5;
then  ( ( ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) or ( ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1, XREAL_1:26;
then  ( x in K0 or x in K1 ) ;
hence  x in K0 \/ K1 by XBOOLE_0:def_3; ::_thesis: verum
end;
A13: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= NonZero (TOP-REAL 2) by A4, PRE_TOPC:def_5 ;
A14: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8;
A15: rng (Sq_Circ | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
proof
reconsider K00 = K0 as Subset of (TOP-REAL 2) by A14, XBOOLE_1:1;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Sq_Circ | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) )
A16: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K00) holds
q `1 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K00) implies q `1 <> 0 )
A17: the carrier of ((TOP-REAL 2) | K00) = K0 by PRE_TOPC:8;
assume  q in the carrier of ((TOP-REAL 2) | K00) ; ::_thesis: q `1 <> 0
then A18: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A17;
now__::_thesis:_not_q_`1_=_0
assume A19: q `1 = 0 ; ::_thesis: contradiction
then  q `2 = 0 by A18;
hence  contradiction by A18, A19, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence  q `1 <> 0 ; ::_thesis: verum
end;
assume  y in rng (Sq_Circ | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0)
then consider x being set  such that
A20: x in dom (Sq_Circ | K0) and
A21: y = (Sq_Circ | K0) . x by FUNCT_1:def_3;
A22: x in (dom Sq_Circ) /\ K0 by A20, RELAT_1:61;
then A23: x in K0 by XBOOLE_0:def_4;
 K0 c= the carrier of (TOP-REAL 2) by A14, XBOOLE_1:1;
then reconsider p = x as Point of (TOP-REAL 2) by A23;
 K00 = the carrier of ((TOP-REAL 2) | K00) by PRE_TOPC:8;
then  p in the carrier of ((TOP-REAL 2) | K00) by A22, XBOOLE_0:def_4;
then A24: p `1 <> 0 by A16;
A25: ex px being Point of (TOP-REAL 2) st
( x = px & ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) by A23;
then A26: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by Def1;
A27: sqrt (1 + (((p `2) / (p `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
then  ( ( (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) & (- (p `1)) / (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ) or ( (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) >= (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) & (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (- (p `1)) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ) ) by A25, XREAL_1:72;
then A28: ( ( (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) & - ((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))) <= (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ) or ( (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) >= (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) & (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) <= - ((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))) ) ) by XCMPLX_1:187;
set p9 = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]|;
A29: ( |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) & |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 = (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) ) by EUCLID:52;
A30: |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52;
A31: now__::_thesis:_not_|[((p_`1)_/_(sqrt_(1_+_(((p_`2)_/_(p_`1))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`2)_/_(p_`1))_^2))))]|_=_0._(TOP-REAL_2)
assume  |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then  0 * (sqrt (1 + (((p `2) / (p `1)) ^2))) = ((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))) * (sqrt (1 + (((p `2) / (p `1)) ^2))) by A30, EUCLID:52, EUCLID:54;
hence  contradiction by A24, A27, XCMPLX_1:87; ::_thesis: verum
end;
 Sq_Circ . p = y by A21, A23, FUNCT_1:49;
then  y in K0 by A31, A26, A28, A29;
then  y in [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5;
hence  y in the carrier of (((TOP-REAL 2) | D) | K0) ; ::_thesis: verum
end;
A32: 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 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
hence  z in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
dom (Sq_Circ | K0) = (dom Sq_Circ) /\ K0 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K0 by FUNCT_2:def_1
.= K0 by A32, XBOOLE_1:28 ;
then reconsider f = Sq_Circ | K0 as Function of (((TOP-REAL 2) | D) | K0),((TOP-REAL 2) | D) by A7, A15, FUNCT_2:2, XBOOLE_1:1;
A33: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5;
A34: 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 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
hence  z in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
A35: rng (Sq_Circ | K1) c= the carrier of (((TOP-REAL 2) | D) | K1)
proof
reconsider K10 = K1 as Subset of (TOP-REAL 2) by A34;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Sq_Circ | K1) or y in the carrier of (((TOP-REAL 2) | D) | K1) )
A36: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K10) holds
q `2 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K10) implies q `2 <> 0 )
A37: the carrier of ((TOP-REAL 2) | K10) = K1 by PRE_TOPC:8;
assume  q in the carrier of ((TOP-REAL 2) | K10) ; ::_thesis: q `2 <> 0
then A38: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A37;
now__::_thesis:_not_q_`2_=_0
assume A39: q `2 = 0 ; ::_thesis: contradiction
then  q `1 = 0 by A38;
hence  contradiction by A38, A39, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence  q `2 <> 0 ; ::_thesis: verum
end;
assume  y in rng (Sq_Circ | K1) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K1)
then consider x being set  such that
A40: x in dom (Sq_Circ | K1) and
A41: y = (Sq_Circ | K1) . x by FUNCT_1:def_3;
A42: x in (dom Sq_Circ) /\ K1 by A40, RELAT_1:61;
then A43: x in K1 by XBOOLE_0:def_4;
then reconsider p = x as Point of (TOP-REAL 2) by A34;
 K10 = the carrier of ((TOP-REAL 2) | K10) by PRE_TOPC:8;
then  p in the carrier of ((TOP-REAL 2) | K10) by A42, XBOOLE_0:def_4;
then A44: p `2 <> 0 by A36;
set p9 = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]|;
A45: ( |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) & |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 = (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ) by EUCLID:52;
A46: ex px being Point of (TOP-REAL 2) st
( x = px & ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) by A43;
then A47: Sq_Circ . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by Th4;
A48: sqrt (1 + (((p `1) / (p `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
then  ( ( (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) & (- (p `2)) / (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ) or ( (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) >= (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) & (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (- (p `2)) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ) ) by A46, XREAL_1:72;
then A49: ( ( (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) & - ((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2)))) <= (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) ) or ( (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) >= (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) & (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) <= - ((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2)))) ) ) by XCMPLX_1:187;
A50: |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52;
A51: now__::_thesis:_not_|[((p_`1)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_/_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_=_0._(TOP-REAL_2)
assume  |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then  0 * (sqrt (1 + (((p `1) / (p `2)) ^2))) = ((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2)))) * (sqrt (1 + (((p `1) / (p `2)) ^2))) by A50, EUCLID:52, EUCLID:54;
hence  contradiction by A44, A48, XCMPLX_1:87; ::_thesis: verum
end;
 Sq_Circ . p = y by A41, A43, FUNCT_1:49;
then  y in K1 by A51, A47, A49, A45;
hence  y in the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; ::_thesis: verum
end;
dom (Sq_Circ | K1) = (dom Sq_Circ) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by FUNCT_2:def_1
.= K1 by A34, XBOOLE_1:28 ;
then reconsider g = Sq_Circ | K1 as Function of (((TOP-REAL 2) | D) | K1),((TOP-REAL 2) | D) by A10, A35, FUNCT_2:2, XBOOLE_1:1;
A52: dom g = K1 by A10, FUNCT_2:def_1;
 g = Sq_Circ | K1 ;
then A53: K1 is closed by A4, Th18;
A54: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5;
A55: 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 A56: x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) ; ::_thesis: f . x = g . x
then  x in K0 by A54, XBOOLE_0:def_4;
then  f . x = Sq_Circ . x by FUNCT_1:49;
hence  f . x = g . x by A33, A56, FUNCT_1:49; ::_thesis: verum
end;
 f = Sq_Circ | K0 ;
then A57: K0 is closed by A4, Th17;
A58: dom f = K0 by A7, FUNCT_2:def_1;
 D = [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5;
then A59: ([#] (((TOP-REAL 2) | D) | K0)) \/ ([#] (((TOP-REAL 2) | D) | K1)) = [#] ((TOP-REAL 2) | D) by A54, A33, A11, XBOOLE_0:def_10;
A60: ( f is continuous & g is continuous ) by A4, Th17, Th18;
then consider h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) such that
A61: h = f +* g and
 h is continuous by A54, A33, A59, A57, A53, A55, JGRAPH_2:1;
 ( K0 = [#] (((TOP-REAL 2) | D) | K0) & K1 = [#] (((TOP-REAL 2) | D) | K1) ) by PRE_TOPC:def_5;
then A62: f tolerates g by A55, A58, A52, PARTFUN1:def_4;
A63: for x being set st x in dom h holds
h . x = (Sq_Circ | D) . x
proof
let x be set ; ::_thesis: ( x in dom h implies h . x = (Sq_Circ | D) . x )
assume A64: x in dom h ; ::_thesis: h . x = (Sq_Circ | D) . x
then reconsider p = x as Point of (TOP-REAL 2) by A13, XBOOLE_0:def_5;
 not x in {(0. (TOP-REAL 2))} by A13, A64, XBOOLE_0:def_5;
then A65: x <> 0. (TOP-REAL 2) by TARSKI:def_1;
 x in the carrier of (TOP-REAL 2) \ (D `) by A3, A13, A64;
then A66: x in (D `) ` by SUBSET_1:def_4;
percases ( x in K0 or not x in K0 ) ;
supposeA67: x in K0 ; ::_thesis: h . x = (Sq_Circ | D) . x
A68: (Sq_Circ | D) . p = Sq_Circ . p by A66, FUNCT_1:49
.= f . p by A67, FUNCT_1:49 ;
h . p = (g +* f) . p by A61, A62, FUNCT_4:34
.= f . p by A58, A67, FUNCT_4:13 ;
hence  h . x = (Sq_Circ | D) . x by A68; ::_thesis: verum
end;
suppose not x in K0 ; ::_thesis: h . x = (Sq_Circ | D) . x
then  ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A65;
then  ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by XREAL_1:26;
then A69: x in K1 by A65;
(Sq_Circ | D) . p = Sq_Circ . p by A66, FUNCT_1:49
.= g . p by A69, FUNCT_1:49 ;
hence  h . x = (Sq_Circ | D) . x by A61, A52, A69, 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 = Sq_Circ | D by A61, A2, A63, FUNCT_1:2;
hence  ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Sq_Circ | D & h is continuous ) by A54, A33, A59, A57, A60, A53, A55, JGRAPH_2:1; ::_thesis: verum
end;

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))}
proof
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( D = NonZero (TOP-REAL 2) implies D ` = {(0. (TOP-REAL 2))} )
assume A1: D = NonZero (TOP-REAL 2) ; ::_thesis: D ` = {(0. (TOP-REAL 2))}
A2: D ` c= {(0. (TOP-REAL 2))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D ` or x in {(0. (TOP-REAL 2))} )
assume A3: x in D ` ; ::_thesis: x in {(0. (TOP-REAL 2))}
then  x in the carrier of (TOP-REAL 2) \ D by SUBSET_1:def_4;
then  not x in D by XBOOLE_0:def_5;
hence  x in {(0. (TOP-REAL 2))} by A1, A3, XBOOLE_0:def_5; ::_thesis: verum
end;
 {(0. (TOP-REAL 2))} c= D `
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(0. (TOP-REAL 2))} or x in D ` )
assume A4: x in {(0. (TOP-REAL 2))} ; ::_thesis: x in D `
then  not x in D by A1, XBOOLE_0:def_5;
then  x in the carrier of (TOP-REAL 2) \ D by A4, XBOOLE_0:def_5;
hence  x in D ` by SUBSET_1:def_4; ::_thesis: verum
end;
hence  D ` = {(0. (TOP-REAL 2))} by A2, XBOOLE_0:def_10; ::_thesis: verum
end;

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 )
proof
reconsider D = NonZero (TOP-REAL 2) as non empty Subset of (TOP-REAL 2) by JGRAPH_2:9;
reconsider f = Sq_Circ as Function of (TOP-REAL 2),(TOP-REAL 2) ;
A1: 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))
A2: [#] ((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 A2, XBOOLE_0:def_5;
then A3: not p = 0. (TOP-REAL 2) by TARSKI:def_1;
percases ( ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) or ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ;
supposeA4: ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2))
then A5: q `2 <> 0 ;
set q9 = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|;
A6: |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52;
A7: sqrt (1 + (((q `1) / (q `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A8: now__::_thesis:_not_|[((q_`1)_/_(sqrt_(1_+_(((q_`1)_/_(q_`2))_^2)))),((q_`2)_/_(sqrt_(1_+_(((q_`1)_/_(q_`2))_^2))))]|_=_0._(TOP-REAL_2)
assume  |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then  0 * (sqrt (1 + (((q `1) / (q `2)) ^2))) = ((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) * (sqrt (1 + (((q `1) / (q `2)) ^2))) by A6, EUCLID:52, EUCLID:54;
hence  contradiction by A5, A7, XCMPLX_1:87; ::_thesis: verum
end;
 Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by A3, A4, Def1;
hence  f . p <> f . (0. (TOP-REAL 2)) by A8, Def1; ::_thesis: verum
end;
supposeA9: ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2))
A10: now__::_thesis:_not_q_`1_=_0
assume A11: q `1 = 0 ; ::_thesis: contradiction
then  q `2 = 0 by A9;
hence  contradiction by A3, A11, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
set q9 = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|;
A12: |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52;
A13: sqrt (1 + (((q `2) / (q `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A14: now__::_thesis:_not_|[((q_`1)_/_(sqrt_(1_+_(((q_`2)_/_(q_`1))_^2)))),((q_`2)_/_(sqrt_(1_+_(((q_`2)_/_(q_`1))_^2))))]|_=_0._(TOP-REAL_2)
assume  |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then  0 * (sqrt (1 + (((q `2) / (q `1)) ^2))) = ((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) * (sqrt (1 + (((q `2) / (q `1)) ^2))) by A12, EUCLID:52, EUCLID:54;
hence  contradiction by A10, A13, XCMPLX_1:87; ::_thesis: verum
end;
 Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by A3, A9, Def1;
hence  f . p <> f . (0. (TOP-REAL 2)) by A14, Def1; ::_thesis: verum
end;
end;
end;
A15: f . (0. (TOP-REAL 2)) = 0. (TOP-REAL 2) by Def1;
A16: 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 Lm16;
assume that
A17: f . (0. (TOP-REAL 2)) in V and
A18: 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 A18, Lm16, PRE_TOPC:30;
then consider r being real number  such that
A19: r > 0 and
A20: Ball (u0,r) c= V by A15, A17, TOPMETR:15;
reconsider r = r as Real by XREAL_0:def_1;
reconsider W1 = Ball (u0,r) as Subset of (TOP-REAL 2) by EUCLID:67;
A21: W1 is open by GOBOARD6:3;
A22: 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
A23: y in dom f and
A24: y in W1 and
A25: z = f . y by FUNCT_1:def_6;
 z in rng f by A23, A25, 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 A23;
reconsider qy = q as Point of (Euclid 2) by EUCLID:67;
 dist (u0,qy) < r by A24, METRIC_1:11;
then  |.((0. (TOP-REAL 2)) - q).| < r by JGRAPH_1:28;
then  sqrt (((((0. (TOP-REAL 2)) - q) `1) ^2) + ((((0. (TOP-REAL 2)) - q) `2) ^2)) < r by JGRAPH_1:30;
then  sqrt (((((0. (TOP-REAL 2)) `1) - (q `1)) ^2) + ((((0. (TOP-REAL 2)) - q) `2) ^2)) < r by TOPREAL3:3;
then A26: sqrt (((((0. (TOP-REAL 2)) `1) - (q `1)) ^2) + ((((0. (TOP-REAL 2)) `2) - (q `2)) ^2)) < r by TOPREAL3:3;
percases ( q = 0. (TOP-REAL 2) or ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) or ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ;
suppose q = 0. (TOP-REAL 2) ; ::_thesis: z in W1
hence  z in W1 by A24, A25, Def1; ::_thesis: verum
end;
supposeA27: ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; ::_thesis: z in W1
A28: (q `2) ^2 >= 0 by XREAL_1:63;
 ((q `2) / (q `1)) ^2 >= 0 by XREAL_1:63;
then  1 + (((q `2) / (q `1)) ^2) >= 1 + 0 by XREAL_1:7;
then A29: sqrt (1 + (((q `2) / (q `1)) ^2)) >= 1 by SQUARE_1:18, SQUARE_1:26;
then  (sqrt (1 + (((q `2) / (q `1)) ^2))) ^2 >= sqrt (1 + (((q `2) / (q `1)) ^2)) by XREAL_1:151;
then A30: 1 <= (sqrt (1 + (((q `2) / (q `1)) ^2))) ^2 by A29, XXREAL_0:2;
A31: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by A27, Def1;
then (qz `2) ^2 = ((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2 by A25, EUCLID:52
.= ((q `2) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2) by XCMPLX_1:76 ;
then A32: (qz `2) ^2 <= ((q `2) ^2) / 1 by A30, A28, XREAL_1:118;
A33: (q `1) ^2 >= 0 by XREAL_1:63;
(qz `1) ^2 = ((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))) ^2 by A25, A31, EUCLID:52
.= ((q `1) ^2) / ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2) by XCMPLX_1:76 ;
then  (qz `1) ^2 <= ((q `1) ^2) / 1 by A30, A33, XREAL_1:118;
then A34: ((qz `1) ^2) + ((qz `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) by A32, XREAL_1:7;
 ( (qz `1) ^2 >= 0 & (qz `2) ^2 >= 0 ) by XREAL_1:63;
then A35: sqrt (((qz `1) ^2) + ((qz `2) ^2)) <= sqrt (((q `1) ^2) + ((q `2) ^2)) by A34, SQUARE_1:26;
A36: ((0. (TOP-REAL 2)) - qz) `2 = ((0. (TOP-REAL 2)) `2) - (qz `2) by TOPREAL3:3
.= - (qz `2) by JGRAPH_2:3 ;
((0. (TOP-REAL 2)) - qz) `1 = ((0. (TOP-REAL 2)) `1) - (qz `1) by TOPREAL3:3
.= - (qz `1) by JGRAPH_2:3 ;
then  sqrt (((((0. (TOP-REAL 2)) - qz) `1) ^2) + ((((0. (TOP-REAL 2)) - qz) `2) ^2)) < r by A26, A36, A35, JGRAPH_2:3, XXREAL_0:2;
then  |.((0. (TOP-REAL 2)) - qz).| < r by JGRAPH_1:30;
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) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: z in W1
A38: (q `2) ^2 >= 0 by XREAL_1:63;
 ((q `1) / (q `2)) ^2 >= 0 by XREAL_1:63;
then  1 + (((q `1) / (q `2)) ^2) >= 1 + 0 by XREAL_1:7;
then A39: sqrt (1 + (((q `1) / (q `2)) ^2)) >= 1 by SQUARE_1:18, SQUARE_1:26;
then  (sqrt (1 + (((q `1) / (q `2)) ^2))) ^2 >= sqrt (1 + (((q `1) / (q `2)) ^2)) by XREAL_1:151;
then A40: 1 <= (sqrt (1 + (((q `1) / (q `2)) ^2))) ^2 by A39, XXREAL_0:2;
A41: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by A37, Def1;
then (qz `2) ^2 = ((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2 by A25, EUCLID:52
.= ((q `2) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2) by XCMPLX_1:76 ;
then A42: (qz `2) ^2 <= ((q `2) ^2) / 1 by A40, A38, XREAL_1:118;
A43: (q `1) ^2 >= 0 by XREAL_1:63;
(qz `1) ^2 = ((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))) ^2 by A25, A41, EUCLID:52
.= ((q `1) ^2) / ((sqrt (1 + (((q `1) / (q `2)) ^2))) ^2) by XCMPLX_1:76 ;
then  (qz `1) ^2 <= ((q `1) ^2) / 1 by A40, A43, XREAL_1:118;
then A44: ((qz `1) ^2) + ((qz `2) ^2) <= ((q `1) ^2) + ((q `2) ^2) by A42, XREAL_1:7;
 ( (qz `1) ^2 >= 0 & (qz `2) ^2 >= 0 ) by XREAL_1:63;
then A45: sqrt (((qz `1) ^2) + ((qz `2) ^2)) <= sqrt (((q `1) ^2) + ((q `2) ^2)) by A44, SQUARE_1:26;
A46: ((0. (TOP-REAL 2)) - qz) `2 = ((0. (TOP-REAL 2)) `2) - (qz `2) by TOPREAL3:3
.= - (qz `2) by JGRAPH_2:3 ;
((0. (TOP-REAL 2)) - qz) `1 = ((0. (TOP-REAL 2)) `1) - (qz `1) by TOPREAL3:3
.= - (qz `1) by JGRAPH_2:3 ;
then  sqrt (((((0. (TOP-REAL 2)) - qz) `1) ^2) + ((((0. (TOP-REAL 2)) - qz) `2) ^2)) < r by A26, A46, A45, JGRAPH_2:3, XXREAL_0:2;
then  |.((0. (TOP-REAL 2)) - qz).| < r by JGRAPH_1:30;
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 A19, 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 A20, A21, A22, XBOOLE_1:1; ::_thesis: verum
end;
A47: D ` = {(0. (TOP-REAL 2))} by Th20;
then  ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Sq_Circ | D & h is continuous ) by Th19;
hence  ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h = Sq_Circ & h is continuous ) by A15, A47, A1, A16, Th3; ::_thesis: verum
end;

theorem Th22: :: JGRAPH_3:22
Sq_Circ is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom Sq_Circ or not x2 in dom Sq_Circ or not Sq_Circ . x1 = Sq_Circ . x2 or x1 = x2 )
assume that
A1: x1 in dom Sq_Circ and
A2: x2 in dom Sq_Circ and
A3: Sq_Circ . x1 = Sq_Circ . x2 ; ::_thesis: x1 = x2
reconsider p2 = x2 as Point of (TOP-REAL 2) by A2;
reconsider p1 = x1 as Point of (TOP-REAL 2) by A1;
set q = p1;
set p = p2;
percases ( p1 = 0. (TOP-REAL 2) or ( p1 <> 0. (TOP-REAL 2) & ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) or ( p1 <> 0. (TOP-REAL 2) & not ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) & not ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) ;
supposeA4: p1 = 0. (TOP-REAL 2) ; ::_thesis: x1 = x2
then A5: Sq_Circ . p1 = 0. (TOP-REAL 2) by Def1;
now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_contradiction_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_contradiction_)_)
percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ;
case p2 = 0. (TOP-REAL 2) ; ::_thesis: x1 = x2
hence  x1 = x2 by A4; ::_thesis: verum
end;
caseA6: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: contradiction
 ((p2 `2) / (p2 `1)) ^2 >= 0 by XREAL_1:63;
then  1 + (((p2 `2) / (p2 `1)) ^2) >= 1 + 0 by XREAL_1:7;
then A7: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) >= 1 by SQUARE_1:18, SQUARE_1:26;
A8: Sq_Circ . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A6, Def1;
then  (p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A3, A5, EUCLID:52, JGRAPH_2:3;
then A9: p2 `2 = 0 * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A7, XCMPLX_1:87
.= 0 ;
 (p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A3, A5, A8, EUCLID:52, JGRAPH_2:3;
then p2 `1 = 0 * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A7, XCMPLX_1:87
.= 0 ;
hence  contradiction by A6, A9, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
caseA10: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: contradiction
 ((p2 `1) / (p2 `2)) ^2 >= 0 by XREAL_1:63;
then  1 + (((p2 `1) / (p2 `2)) ^2) >= 1 + 0 by XREAL_1:7;
then A11: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) >= 1 by SQUARE_1:18, SQUARE_1:26;
 Sq_Circ . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| by A10, Def1;
then  (p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A3, A5, EUCLID:52, JGRAPH_2:3;
then p2 `2 = 0 * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A11, XCMPLX_1:87
.= 0 ;
hence  contradiction by A10; ::_thesis: verum
end;
end;
end;
hence  x1 = x2 ; ::_thesis: verum
end;
supposeA12: ( p1 <> 0. (TOP-REAL 2) & ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) ; ::_thesis: x1 = x2
A13: sqrt (1 + (((p1 `2) / (p1 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A14: Sq_Circ . p1 = |[((p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| by A12, Def1;
A15: |[((p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| `2 = (p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by EUCLID:52;
A16: 1 + (((p1 `2) / (p1 `1)) ^2) > 0 by Lm1;
A17: |[((p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| `1 = (p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by EUCLID:52;
now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_x1_=_x2_)_)
percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ;
caseA18: p2 = 0. (TOP-REAL 2) ; ::_thesis: contradiction
 ((p1 `2) / (p1 `1)) ^2 >= 0 by XREAL_1:63;
then  1 + (((p1 `2) / (p1 `1)) ^2) >= 1 + 0 by XREAL_1:7;
then A19: sqrt (1 + (((p1 `2) / (p1 `1)) ^2)) >= 1 by SQUARE_1:18, SQUARE_1:26;
A20: Sq_Circ . p2 = 0. (TOP-REAL 2) by A18, Def1;
then  (p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) = 0 by A3, A14, EUCLID:52, JGRAPH_2:3;
then A21: p1 `2 = 0 * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A19, XCMPLX_1:87
.= 0 ;
 (p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) = 0 by A3, A14, A20, EUCLID:52, JGRAPH_2:3;
then p1 `1 = 0 * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A19, XCMPLX_1:87
.= 0 ;
hence  contradiction by A12, A21, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
caseA22: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: x1 = x2
now__::_thesis:_not_p2_`1_=_0
assume A23: p2 `1 = 0 ; ::_thesis: contradiction
then  p2 `2 = 0 by A22;
hence  contradiction by A22, A23, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
then A24: (p2 `1) ^2 > 0 by SQUARE_1:12;
A25: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A26: 1 + (((p2 `2) / (p2 `1)) ^2) > 0 by Lm1;
A27: Sq_Circ . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A22, Def1;
then A28: (p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = (p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A3, A14, A15, EUCLID:52;
then  ((p2 `2) ^2) / ((sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ^2) = ((p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))) ^2 by XCMPLX_1:76;
then  ((p2 `2) ^2) / ((sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ^2) = ((p1 `2) ^2) / ((sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ^2) by XCMPLX_1:76;
then  ((p2 `2) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2)) = ((p1 `2) ^2) / ((sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ^2) by A26, SQUARE_1:def_2;
then A29: ((p2 `2) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2)) = ((p1 `2) ^2) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A16, SQUARE_1:def_2;
A30: (p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = (p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A3, A14, A17, A27, EUCLID:52;
then  ((p2 `1) ^2) / ((sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ^2) = ((p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))) ^2 by XCMPLX_1:76;
then  ((p2 `1) ^2) / ((sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ^2) = ((p1 `1) ^2) / ((sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ^2) by XCMPLX_1:76;
then  ((p2 `1) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2)) = ((p1 `1) ^2) / ((sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ^2) by A26, SQUARE_1:def_2;
then  ((p2 `1) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2)) = ((p1 `1) ^2) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A16, SQUARE_1:def_2;
then  (((p2 `1) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2))) / ((p2 `1) ^2) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by XCMPLX_1:48;
then  (((p2 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p2 `2) / (p2 `1)) ^2)) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by XCMPLX_1:48;
then  1 / (1 + (((p2 `2) / (p2 `1)) ^2)) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A24, XCMPLX_1:60;
then A31: (1 / (1 + (((p2 `2) / (p2 `1)) ^2))) * (1 + (((p1 `2) / (p1 `1)) ^2)) = ((p1 `1) ^2) / ((p2 `1) ^2) by A16, XCMPLX_1:87;
now__::_thesis:_not_p1_`1_=_0
assume A32: p1 `1 = 0 ; ::_thesis: contradiction
then  p1 `2 = 0 by A12;
hence  contradiction by A12, A32, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
then A33: (p1 `1) ^2 > 0 by SQUARE_1:12;
now__::_thesis:_(_(_p2_`2_=_0_&_x1_=_x2_)_or_(_p2_`2_<>_0_&_x1_=_x2_)_)
percases ( p2 `2 = 0 or p2 `2 <> 0 ) ;
caseA34: p2 `2 = 0 ; ::_thesis: x1 = x2
then  (p1 `2) ^2 = 0 by A16, A29, XCMPLX_1:50;
then A35: p1 `2 = 0 by XCMPLX_1:6;
then  p2 = |[(p1 `1),0]| by A3, A14, A27, A34, EUCLID:53, SQUARE_1:18;
hence  x1 = x2 by A35, EUCLID:53; ::_thesis: verum
end;
case p2 `2 <> 0 ; ::_thesis: x1 = x2
then A36: (p2 `2) ^2 > 0 by SQUARE_1:12;
 (((p2 `2) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2))) / ((p2 `2) ^2) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A29, XCMPLX_1:48;
then  (((p2 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p2 `2) / (p2 `1)) ^2)) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by XCMPLX_1:48;
then  1 / (1 + (((p2 `2) / (p2 `1)) ^2)) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A36, XCMPLX_1:60;
then  (1 / (1 + (((p2 `2) / (p2 `1)) ^2))) * (1 + (((p1 `2) / (p1 `1)) ^2)) = ((p1 `2) ^2) / ((p2 `2) ^2) by A16, XCMPLX_1:87;
then  (((p1 `1) ^2) / ((p1 `1) ^2)) / ((p2 `1) ^2) = (((p1 `2) ^2) / ((p2 `2) ^2)) / ((p1 `1) ^2) by A31, XCMPLX_1:48;
then  1 / ((p2 `1) ^2) = (((p1 `2) ^2) / ((p2 `2) ^2)) / ((p1 `1) ^2) by A33, XCMPLX_1:60;
then  (1 / ((p2 `1) ^2)) * ((p2 `2) ^2) = (((p2 `2) ^2) * (((p1 `2) ^2) / ((p2 `2) ^2))) / ((p1 `1) ^2) by XCMPLX_1:74;
then  (1 / ((p2 `1) ^2)) * ((p2 `2) ^2) = ((p1 `2) ^2) / ((p1 `1) ^2) by A36, XCMPLX_1:87;
then  ((p2 `2) ^2) / ((p2 `1) ^2) = ((p1 `2) ^2) / ((p1 `1) ^2) by XCMPLX_1:99;
then  ((p2 `2) / (p2 `1)) ^2 = ((p1 `2) ^2) / ((p1 `1) ^2) by XCMPLX_1:76;
then A37: 1 + (((p2 `2) / (p2 `1)) ^2) = 1 + (((p1 `2) / (p1 `1)) ^2) by XCMPLX_1:76;
then  p2 `2 = ((p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A28, A25, XCMPLX_1:87;
then A38: p2 `2 = p1 `2 by A13, XCMPLX_1:87;
 p2 `1 = ((p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A30, A25, A37, XCMPLX_1:87;
then  p2 `1 = p1 `1 by A13, XCMPLX_1:87;
then  p2 = |[(p1 `1),(p1 `2)]| by A38, EUCLID:53;
hence  x1 = x2 by EUCLID:53; ::_thesis: verum
end;
end;
end;
hence  x1 = x2 ; ::_thesis: verum
end;
caseA39: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: x1 = x2
A40: 1 + (((p2 `1) / (p2 `2)) ^2) > 0 by Lm1;
A41: ( ( p2 <> 0. (TOP-REAL 2) & p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A39, JGRAPH_2:13;
 p2 `2 <> 0 by A39;
then A42: (p2 `2) ^2 > 0 by SQUARE_1:12;
 |[((p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = (p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52;
then A43: (p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = (p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A3, A14, A15, A39, Def1;
then  ((p2 `2) ^2) / ((sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ^2) = ((p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))) ^2 by XCMPLX_1:76;
then  ((p2 `2) ^2) / ((sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ^2) = ((p1 `2) ^2) / ((sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ^2) by XCMPLX_1:76;
then  ((p2 `2) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2)) = ((p1 `2) ^2) / ((sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ^2) by A40, SQUARE_1:def_2;
then  ((p2 `2) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2)) = ((p1 `2) ^2) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A16, SQUARE_1:def_2;
then  (((p2 `2) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2))) / ((p2 `2) ^2) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by XCMPLX_1:48;
then  (((p2 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p2 `1) / (p2 `2)) ^2)) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by XCMPLX_1:48;
then  1 / (1 + (((p2 `1) / (p2 `2)) ^2)) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A42, XCMPLX_1:60;
then A44: (1 / (1 + (((p2 `1) / (p2 `2)) ^2))) * (1 + (((p1 `2) / (p1 `1)) ^2)) = ((p1 `2) ^2) / ((p2 `2) ^2) by A16, XCMPLX_1:87;
A45: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
 |[((p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = (p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52;
then A46: (p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = (p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A3, A14, A17, A39, Def1;
then  ((p2 `1) ^2) / ((sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ^2) = ((p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))) ^2 by XCMPLX_1:76;
then  ((p2 `1) ^2) / ((sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ^2) = ((p1 `1) ^2) / ((sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ^2) by XCMPLX_1:76;
then  ((p2 `1) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2)) = ((p1 `1) ^2) / ((sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ^2) by A40, SQUARE_1:def_2;
then A47: ((p2 `1) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2)) = ((p1 `1) ^2) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A16, SQUARE_1:def_2;
A48: now__::_thesis:_not_p1_`1_=_0
assume A49: p1 `1 = 0 ; ::_thesis: contradiction
then  p1 `2 = 0 by A12;
hence  contradiction by A12, A49, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
then A50: (p1 `1) ^2 > 0 by SQUARE_1:12;
now__::_thesis:_(_(_p2_`1_=_0_&_contradiction_)_or_(_p2_`1_<>_0_&_x1_=_x2_)_)
percases ( p2 `1 = 0 or p2 `1 <> 0 ) ;
case p2 `1 = 0 ; ::_thesis: contradiction
then  (p1 `1) ^2 = 0 by A16, A47, XCMPLX_1:50;
then A51: p1 `1 = 0 by XCMPLX_1:6;
then  p1 `2 = 0 by A12;
hence  contradiction by A12, A51, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
caseA52: p2 `1 <> 0 ; ::_thesis: x1 = x2
set a = (p1 `2) / (p1 `1);
 (((p2 `1) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2))) / ((p2 `1) ^2) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A47, XCMPLX_1:48;
then A53: (((p2 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p2 `1) / (p2 `2)) ^2)) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by XCMPLX_1:48;
A54: ( ( (p1 `1) * ((p1 `2) / (p1 `1)) <= p1 `1 & - (p1 `1) <= (p1 `1) * ((p1 `2) / (p1 `1)) ) or ( (p1 `1) * ((p1 `2) / (p1 `1)) >= p1 `1 & (p1 `1) * ((p1 `2) / (p1 `1)) <= - (p1 `1) ) ) by A12, A48, XCMPLX_1:87;
A55: now__::_thesis:_(_(_p1_`1_>_0_&_(_(_(p1_`2)_/_(p1_`1)_<=_1_&_-_1_<=_(p1_`2)_/_(p1_`1)_)_or_(_(p1_`2)_/_(p1_`1)_>=_1_&_(p1_`2)_/_(p1_`1)_<=_-_1_)_)_)_or_(_p1_`1_<_0_&_(_(_(p1_`2)_/_(p1_`1)_<=_1_&_-_1_<=_(p1_`2)_/_(p1_`1)_)_or_(_(p1_`2)_/_(p1_`1)_>=_1_&_(p1_`2)_/_(p1_`1)_<=_-_1_)_)_)_)
percases ( p1 `1 > 0 or p1 `1 < 0 ) by A48;
caseA56: p1 `1 > 0 ; ::_thesis: ( ( (p1 `2) / (p1 `1) <= 1 & - 1 <= (p1 `2) / (p1 `1) ) or ( (p1 `2) / (p1 `1) >= 1 & (p1 `2) / (p1 `1) <= - 1 ) )
then  ( ( (((p1 `2) / (p1 `1)) * (p1 `1)) / (p1 `1) <= (p1 `1) / (p1 `1) & (- (p1 `1)) / (p1 `1) <= (((p1 `2) / (p1 `1)) * (p1 `1)) / (p1 `1) ) or ( (((p1 `2) / (p1 `1)) * (p1 `1)) / (p1 `1) >= (p1 `1) / (p1 `1) & (((p1 `2) / (p1 `1)) * (p1 `1)) / (p1 `1) <= (- (p1 `1)) / (p1 `1) ) ) by A54, XREAL_1:72;
then A57: ( ( (p1 `2) / (p1 `1) <= (p1 `1) / (p1 `1) & (- (p1 `1)) / (p1 `1) <= (p1 `2) / (p1 `1) ) or ( (p1 `2) / (p1 `1) >= (p1 `1) / (p1 `1) & (p1 `2) / (p1 `1) <= (- (p1 `1)) / (p1 `1) ) ) by A56, XCMPLX_1:89;
 (p1 `1) / (p1 `1) = 1 by A56, XCMPLX_1:60;
hence  ( ( (p1 `2) / (p1 `1) <= 1 & - 1 <= (p1 `2) / (p1 `1) ) or ( (p1 `2) / (p1 `1) >= 1 & (p1 `2) / (p1 `1) <= - 1 ) ) by A57, XCMPLX_1:187; ::_thesis: verum
end;
caseA58: p1 `1 < 0 ; ::_thesis: ( ( (p1 `2) / (p1 `1) <= 1 & - 1 <= (p1 `2) / (p1 `1) ) or ( (p1 `2) / (p1 `1) >= 1 & (p1 `2) / (p1 `1) <= - 1 ) )
then A59: ( (p1 `1) / (p1 `1) = 1 & (- (p1 `1)) / (p1 `1) = - 1 ) by XCMPLX_1:60, XCMPLX_1:197;
 ( ( (((p1 `2) / (p1 `1)) * (p1 `1)) / (p1 `1) >= (p1 `1) / (p1 `1) & (- (p1 `1)) / (p1 `1) >= (((p1 `2) / (p1 `1)) * (p1 `1)) / (p1 `1) ) or ( (((p1 `2) / (p1 `1)) * (p1 `1)) / (p1 `1) <= (p1 `1) / (p1 `1) & (((p1 `2) / (p1 `1)) * (p1 `1)) / (p1 `1) >= (- (p1 `1)) / (p1 `1) ) ) by A54, A58, XREAL_1:73;
hence  ( ( (p1 `2) / (p1 `1) <= 1 & - 1 <= (p1 `2) / (p1 `1) ) or ( (p1 `2) / (p1 `1) >= 1 & (p1 `2) / (p1 `1) <= - 1 ) ) by A58, A59, XCMPLX_1:89; ::_thesis: verum
end;
end;
end;
 (p2 `1) ^2 > 0 by A52, SQUARE_1:12;
then  1 / (1 + (((p2 `1) / (p2 `2)) ^2)) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `2) / (p1 `1)) ^2)) by A53, XCMPLX_1:60;
then  (1 / (1 + (((p2 `1) / (p2 `2)) ^2))) * (1 + (((p1 `2) / (p1 `1)) ^2)) = ((p1 `1) ^2) / ((p2 `1) ^2) by A16, XCMPLX_1:87;
then  (((p1 `1) ^2) / ((p1 `1) ^2)) / ((p2 `1) ^2) = (((p1 `2) ^2) / ((p2 `2) ^2)) / ((p1 `1) ^2) by A44, XCMPLX_1:48;
then  1 / ((p2 `1) ^2) = (((p1 `2) ^2) / ((p2 `2) ^2)) / ((p1 `1) ^2) by A50, XCMPLX_1:60;
then  (1 / ((p2 `1) ^2)) * ((p2 `2) ^2) = (((p2 `2) ^2) * (((p1 `2) ^2) / ((p2 `2) ^2))) / ((p1 `1) ^2) by XCMPLX_1:74;
then  (1 / ((p2 `1) ^2)) * ((p2 `2) ^2) = ((p1 `2) ^2) / ((p1 `1) ^2) by A42, XCMPLX_1:87;
then  ((p2 `2) ^2) / ((p2 `1) ^2) = ((p1 `2) ^2) / ((p1 `1) ^2) by XCMPLX_1:99;
then  ((p2 `2) / (p2 `1)) ^2 = ((p1 `2) ^2) / ((p1 `1) ^2) by XCMPLX_1:76;
then A60: ((p2 `2) / (p2 `1)) ^2 = ((p1 `2) / (p1 `1)) ^2 by XCMPLX_1:76;
then A61: ( ((p2 `2) / (p2 `1)) * (p2 `1) = ((p1 `2) / (p1 `1)) * (p2 `1) or ((p2 `2) / (p2 `1)) * (p2 `1) = (- ((p1 `2) / (p1 `1))) * (p2 `1) ) by SQUARE_1:40;
A62: now__::_thesis:_(_(_p2_`2_=_((p1_`2)_/_(p1_`1))_*_(p2_`1)_&_(_1_<=_(p1_`2)_/_(p1_`1)_or_-_1_>=_(p1_`2)_/_(p1_`1)_)_)_or_(_p2_`2_=_(-_((p1_`2)_/_(p1_`1)))_*_(p2_`1)_&_(_1_<=_(p1_`2)_/_(p1_`1)_or_-_1_>=_(p1_`2)_/_(p1_`1)_)_)_)
percases ( p2 `2 = ((p1 `2) / (p1 `1)) * (p2 `1) or p2 `2 = (- ((p1 `2) / (p1 `1))) * (p2 `1) ) by A52, A61, XCMPLX_1:87;
caseA63: p2 `2 = ((p1 `2) / (p1 `1)) * (p2 `1) ; ::_thesis: ( 1 <= (p1 `2) / (p1 `1) or - 1 >= (p1 `2) / (p1 `1) )
now__::_thesis:_(_(_p2_`1_>_0_&_(_(_1_<=_(p1_`2)_/_(p1_`1)_&_-_((p1_`2)_/_(p1_`1))_<=_1_)_or_(_1_>=_(p1_`2)_/_(p1_`1)_&_1_<=_-_((p1_`2)_/_(p1_`1))_)_)_)_or_(_p2_`1_<_0_&_(_(_1_<=_(p1_`2)_/_(p1_`1)_&_-_((p1_`2)_/_(p1_`1))_<=_1_)_or_(_1_>=_(p1_`2)_/_(p1_`1)_&_1_<=_-_((p1_`2)_/_(p1_`1))_)_)_)_)
percases ( p2 `1 > 0 or p2 `1 < 0 ) by A52;
case p2 `1 > 0 ; ::_thesis: ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & 1 <= - ((p1 `2) / (p1 `1)) ) )
then  ( ( (p2 `1) / (p2 `1) <= (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) & (- (((p1 `2) / (p1 `1)) * (p2 `1))) / (p2 `1) <= (p2 `1) / (p2 `1) ) or ( (p2 `1) / (p2 `1) >= (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) & (p2 `1) / (p2 `1) <= (- (((p1 `2) / (p1 `1)) * (p2 `1))) / (p2 `1) ) ) by A41, A63, XREAL_1:72;
then A64: ( ( 1 <= (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) & (- (((p1 `2) / (p1 `1)) * (p2 `1))) / (p2 `1) <= 1 ) or ( 1 >= (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) & 1 <= (- (((p1 `2) / (p1 `1)) * (p2 `1))) / (p2 `1) ) ) by A52, XCMPLX_1:60;
 (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) = (p1 `2) / (p1 `1) by A52, XCMPLX_1:89;
hence  ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & 1 <= - ((p1 `2) / (p1 `1)) ) ) by A64, XCMPLX_1:187; ::_thesis: verum
end;
case p2 `1 < 0 ; ::_thesis: ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & 1 <= - ((p1 `2) / (p1 `1)) ) )
then  ( ( (p2 `1) / (p2 `1) >= (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) & (- (((p1 `2) / (p1 `1)) * (p2 `1))) / (p2 `1) >= (p2 `1) / (p2 `1) ) or ( (p2 `1) / (p2 `1) <= (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) & (p2 `1) / (p2 `1) >= (- (((p1 `2) / (p1 `1)) * (p2 `1))) / (p2 `1) ) ) by A41, A63, XREAL_1:73;
then A65: ( ( 1 >= (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) & (- (((p1 `2) / (p1 `1)) * (p2 `1))) / (p2 `1) >= 1 ) or ( 1 <= (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) & 1 >= (- (((p1 `2) / (p1 `1)) * (p2 `1))) / (p2 `1) ) ) by A52, XCMPLX_1:60;
 (((p1 `2) / (p1 `1)) * (p2 `1)) / (p2 `1) = (p1 `2) / (p1 `1) by A52, XCMPLX_1:89;
hence  ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & 1 <= - ((p1 `2) / (p1 `1)) ) ) by A65, XCMPLX_1:187; ::_thesis: verum
end;
end;
end;
then  ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & - 1 >= - (- ((p1 `2) / (p1 `1))) ) ) by XREAL_1:24;
hence  ( 1 <= (p1 `2) / (p1 `1) or - 1 >= (p1 `2) / (p1 `1) ) ; ::_thesis: verum
end;
caseA66: p2 `2 = (- ((p1 `2) / (p1 `1))) * (p2 `1) ; ::_thesis: ( 1 <= (p1 `2) / (p1 `1) or - 1 >= (p1 `2) / (p1 `1) )
now__::_thesis:_(_(_p2_`1_>_0_&_(_(_1_<=_(p1_`2)_/_(p1_`1)_&_-_((p1_`2)_/_(p1_`1))_<=_1_)_or_(_1_>=_(p1_`2)_/_(p1_`1)_&_1_<=_-_((p1_`2)_/_(p1_`1))_)_)_)_or_(_p2_`1_<_0_&_(_(_1_<=_(p1_`2)_/_(p1_`1)_&_-_((p1_`2)_/_(p1_`1))_<=_1_)_or_(_1_>=_(p1_`2)_/_(p1_`1)_&_1_<=_-_((p1_`2)_/_(p1_`1))_)_)_)_)
percases ( p2 `1 > 0 or p2 `1 < 0 ) by A52;
case p2 `1 > 0 ; ::_thesis: ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & 1 <= - ((p1 `2) / (p1 `1)) ) )
then  ( ( (p2 `1) / (p2 `1) <= ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) & (- ((- ((p1 `2) / (p1 `1))) * (p2 `1))) / (p2 `1) <= (p2 `1) / (p2 `1) ) or ( (p2 `1) / (p2 `1) >= ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) & (p2 `1) / (p2 `1) <= (- ((- ((p1 `2) / (p1 `1))) * (p2 `1))) / (p2 `1) ) ) by A41, A66, XREAL_1:72;
then  ( ( 1 <= ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) & (- ((- ((p1 `2) / (p1 `1))) * (p2 `1))) / (p2 `1) <= 1 ) or ( 1 >= ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) & 1 <= (- ((- ((p1 `2) / (p1 `1))) * (p2 `1))) / (p2 `1) ) ) by A52, XCMPLX_1:60;
then A67: ( ( 1 <= - ((p1 `2) / (p1 `1)) & - (((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1)) <= 1 ) or ( 1 >= - ((p1 `2) / (p1 `1)) & 1 <= - (((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1)) ) ) by A52, XCMPLX_1:89, XCMPLX_1:187;
 ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) = - ((p1 `2) / (p1 `1)) by A52, XCMPLX_1:89;
hence  ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & 1 <= - ((p1 `2) / (p1 `1)) ) ) by A67; ::_thesis: verum
end;
case p2 `1 < 0 ; ::_thesis: ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & 1 <= - ((p1 `2) / (p1 `1)) ) )
then  ( ( (p2 `1) / (p2 `1) >= ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) & (- ((- ((p1 `2) / (p1 `1))) * (p2 `1))) / (p2 `1) >= (p2 `1) / (p2 `1) ) or ( (p2 `1) / (p2 `1) <= ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) & (p2 `1) / (p2 `1) >= (- ((- ((p1 `2) / (p1 `1))) * (p2 `1))) / (p2 `1) ) ) by A41, A66, XREAL_1:73;
then  ( ( 1 >= ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) & (- ((- ((p1 `2) / (p1 `1))) * (p2 `1))) / (p2 `1) >= 1 ) or ( 1 <= ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) & 1 >= (- ((- ((p1 `2) / (p1 `1))) * (p2 `1))) / (p2 `1) ) ) by A52, XCMPLX_1:60;
then A68: ( ( 1 >= - ((p1 `2) / (p1 `1)) & - (((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1)) >= 1 ) or ( 1 <= - ((p1 `2) / (p1 `1)) & 1 >= - (((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1)) ) ) by A52, XCMPLX_1:89, XCMPLX_1:187;
 ((- ((p1 `2) / (p1 `1))) * (p2 `1)) / (p2 `1) = - ((p1 `2) / (p1 `1)) by A52, XCMPLX_1:89;
hence  ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & 1 <= - ((p1 `2) / (p1 `1)) ) ) by A68; ::_thesis: verum
end;
end;
end;
then  ( ( 1 <= (p1 `2) / (p1 `1) & - ((p1 `2) / (p1 `1)) <= 1 ) or ( 1 >= (p1 `2) / (p1 `1) & - 1 >= - (- ((p1 `2) / (p1 `1))) ) ) by XREAL_1:24;
hence  ( 1 <= (p1 `2) / (p1 `1) or - 1 >= (p1 `2) / (p1 `1) ) ; ::_thesis: verum
end;
end;
end;
A69: now__::_thesis:_(_(_(p1_`2)_/_(p1_`1)_=_1_&_((p2_`1)_/_(p2_`2))_^2_=_((p1_`2)_/_(p1_`1))_^2_)_or_(_(p1_`2)_/_(p1_`1)_=_-_1_&_((p2_`1)_/_(p2_`2))_^2_=_((p1_`2)_/_(p1_`1))_^2_)_)
percases ( (p1 `2) / (p1 `1) = 1 or (p1 `2) / (p1 `1) = - 1 ) by A62, A55, XXREAL_0:1;
case (p1 `2) / (p1 `1) = 1 ; ::_thesis: ((p2 `1) / (p2 `2)) ^2 = ((p1 `2) / (p1 `1)) ^2
then  ((p2 `2) ^2) / ((p2 `1) ^2) = 1 by A60, XCMPLX_1:76;
then A70: (p2 `2) ^2 = (p2 `1) ^2 by XCMPLX_1:58;
 ((p2 `1) / (p2 `2)) ^2 = ((p2 `1) ^2) / ((p2 `2) ^2) by XCMPLX_1:76;
hence  ((p2 `1) / (p2 `2)) ^2 = ((p1 `2) / (p1 `1)) ^2 by A60, A70, XCMPLX_1:76; ::_thesis: verum
end;
case (p1 `2) / (p1 `1) = - 1 ; ::_thesis: ((p2 `1) / (p2 `2)) ^2 = ((p1 `2) / (p1 `1)) ^2
then  ((p2 `2) ^2) / ((p2 `1) ^2) = 1 by A60, XCMPLX_1:76;
then A71: (p2 `2) ^2 = (p2 `1) ^2 by XCMPLX_1:58;
 ((p2 `1) / (p2 `2)) ^2 = ((p2 `1) ^2) / ((p2 `2) ^2) by XCMPLX_1:76;
hence  ((p2 `1) / (p2 `2)) ^2 = ((p1 `2) / (p1 `1)) ^2 by A60, A71, XCMPLX_1:76; ::_thesis: verum
end;
end;
end;
then  p2 `2 = ((p1 `2) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A43, A45, XCMPLX_1:87;
then A72: p2 `2 = p1 `2 by A13, XCMPLX_1:87;
 p2 `1 = ((p1 `1) / (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A46, A45, A69, XCMPLX_1:87;
then  p2 `1 = p1 `1 by A13, XCMPLX_1:87;
then  p2 = |[(p1 `1),(p1 `2)]| by A72, EUCLID:53;
hence  x1 = x2 by EUCLID:53; ::_thesis: verum
end;
end;
end;
hence  x1 = x2 ; ::_thesis: verum
end;
end;
end;
hence  x1 = x2 ; ::_thesis: verum
end;
supposeA73: ( p1 <> 0. (TOP-REAL 2) & not ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) & not ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ; ::_thesis: x1 = x2
A74: |[((p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))),((p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))))]| `2 = (p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by EUCLID:52;
A75: |[((p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))),((p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))))]| `1 = (p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by EUCLID:52;
A76: 1 + (((p1 `1) / (p1 `2)) ^2) > 0 by Lm1;
A77: sqrt (1 + (((p1 `1) / (p1 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A78: Sq_Circ . p1 = |[((p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))),((p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))))]| by A73, Def1;
A79: ( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ) by A73, JGRAPH_2:13;
now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_x1_=_x2_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_x1_=_x2_)_)
percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ;
caseA80: p2 = 0. (TOP-REAL 2) ; ::_thesis: contradiction
 ((p1 `1) / (p1 `2)) ^2 >= 0 by XREAL_1:63;
then  1 + (((p1 `1) / (p1 `2)) ^2) >= 1 + 0 by XREAL_1:7;
then A81: sqrt (1 + (((p1 `1) / (p1 `2)) ^2)) >= 1 by SQUARE_1:18, SQUARE_1:26;
 Sq_Circ . p2 = 0. (TOP-REAL 2) by A80, Def1;
then  (p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) = 0 by A3, A78, EUCLID:52, JGRAPH_2:3;
then p1 `2 = 0 * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A81, XCMPLX_1:87
.= 0 ;
hence  contradiction by A73; ::_thesis: verum
end;
caseA82: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: x1 = x2
now__::_thesis:_not_p2_`1_=_0
assume A83: p2 `1 = 0 ; ::_thesis: contradiction
then  p2 `2 = 0 by A82;
hence  contradiction by A82, A83, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
then A84: (p2 `1) ^2 > 0 by SQUARE_1:12;
A85: 1 + (((p2 `2) / (p2 `1)) ^2) > 0 by Lm1;
A86: Sq_Circ . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A82, Def1;
then A87: (p2 `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = (p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A3, A78, A75, EUCLID:52;
then  ((p2 `1) ^2) / ((sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ^2) = ((p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))) ^2 by XCMPLX_1:76;
then  ((p2 `1) ^2) / ((sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ^2) = ((p1 `1) ^2) / ((sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ^2) by XCMPLX_1:76;
then  ((p2 `1) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2)) = ((p1 `1) ^2) / ((sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ^2) by A85, SQUARE_1:def_2;
then  ((p2 `1) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2)) = ((p1 `1) ^2) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A76, SQUARE_1:def_2;
then  (((p2 `1) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2))) / ((p2 `1) ^2) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by XCMPLX_1:48;
then  (((p2 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p2 `2) / (p2 `1)) ^2)) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by XCMPLX_1:48;
then  1 / (1 + (((p2 `2) / (p2 `1)) ^2)) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A84, XCMPLX_1:60;
then A88: (1 / (1 + (((p2 `2) / (p2 `1)) ^2))) * (1 + (((p1 `1) / (p1 `2)) ^2)) = ((p1 `1) ^2) / ((p2 `1) ^2) by A76, XCMPLX_1:87;
A89: (p2 `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = (p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A3, A78, A74, A86, EUCLID:52;
then  ((p2 `2) ^2) / ((sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ^2) = ((p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))) ^2 by XCMPLX_1:76;
then  ((p2 `2) ^2) / ((sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ^2) = ((p1 `2) ^2) / ((sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ^2) by XCMPLX_1:76;
then  ((p2 `2) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2)) = ((p1 `2) ^2) / ((sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ^2) by A85, SQUARE_1:def_2;
then A90: ((p2 `2) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2)) = ((p1 `2) ^2) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A76, SQUARE_1:def_2;
A91: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A92: p1 `2 <> 0 by A73;
then A93: (p1 `2) ^2 > 0 by SQUARE_1:12;
now__::_thesis:_(_(_p2_`2_=_0_&_contradiction_)_or_(_p2_`2_<>_0_&_x1_=_x2_)_)
percases ( p2 `2 = 0 or p2 `2 <> 0 ) ;
case p2 `2 = 0 ; ::_thesis: contradiction
then  (p1 `2) ^2 = 0 by A76, A90, XCMPLX_1:50;
then  p1 `2 = 0 by XCMPLX_1:6;
hence  contradiction by A73; ::_thesis: verum
end;
caseA94: p2 `2 <> 0 ; ::_thesis: x1 = x2
set a = (p1 `1) / (p1 `2);
 (((p2 `2) ^2) / (1 + (((p2 `2) / (p2 `1)) ^2))) / ((p2 `2) ^2) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A90, XCMPLX_1:48;
then A95: (((p2 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p2 `2) / (p2 `1)) ^2)) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by XCMPLX_1:48;
A96: ( ( (p1 `2) * ((p1 `1) / (p1 `2)) <= p1 `2 & - (p1 `2) <= (p1 `2) * ((p1 `1) / (p1 `2)) ) or ( (p1 `2) * ((p1 `1) / (p1 `2)) >= p1 `2 & (p1 `2) * ((p1 `1) / (p1 `2)) <= - (p1 `2) ) ) by A79, A92, XCMPLX_1:87;
A97: now__::_thesis:_(_(_p1_`2_>_0_&_(_(_(p1_`1)_/_(p1_`2)_<=_1_&_-_1_<=_(p1_`1)_/_(p1_`2)_)_or_(_(p1_`1)_/_(p1_`2)_>=_1_&_(p1_`1)_/_(p1_`2)_<=_-_1_)_)_)_or_(_p1_`2_<_0_&_(_(_(p1_`1)_/_(p1_`2)_<=_1_&_-_1_<=_(p1_`1)_/_(p1_`2)_)_or_(_(p1_`1)_/_(p1_`2)_>=_1_&_(p1_`1)_/_(p1_`2)_<=_-_1_)_)_)_)
percases ( p1 `2 > 0 or p1 `2 < 0 ) by A73;
caseA98: p1 `2 > 0 ; ::_thesis: ( ( (p1 `1) / (p1 `2) <= 1 & - 1 <= (p1 `1) / (p1 `2) ) or ( (p1 `1) / (p1 `2) >= 1 & (p1 `1) / (p1 `2) <= - 1 ) )
then A99: ( (p1 `2) / (p1 `2) = 1 & (- (p1 `2)) / (p1 `2) = - 1 ) by XCMPLX_1:60, XCMPLX_1:197;
 ( ( (((p1 `1) / (p1 `2)) * (p1 `2)) / (p1 `2) <= (p1 `2) / (p1 `2) & (- (p1 `2)) / (p1 `2) <= (((p1 `1) / (p1 `2)) * (p1 `2)) / (p1 `2) ) or ( (((p1 `1) / (p1 `2)) * (p1 `2)) / (p1 `2) >= (p1 `2) / (p1 `2) & (((p1 `1) / (p1 `2)) * (p1 `2)) / (p1 `2) <= (- (p1 `2)) / (p1 `2) ) ) by A96, A98, XREAL_1:72;
hence  ( ( (p1 `1) / (p1 `2) <= 1 & - 1 <= (p1 `1) / (p1 `2) ) or ( (p1 `1) / (p1 `2) >= 1 & (p1 `1) / (p1 `2) <= - 1 ) ) by A98, A99, XCMPLX_1:89; ::_thesis: verum
end;
caseA100: p1 `2 < 0 ; ::_thesis: ( ( (p1 `1) / (p1 `2) <= 1 & - 1 <= (p1 `1) / (p1 `2) ) or ( (p1 `1) / (p1 `2) >= 1 & (p1 `1) / (p1 `2) <= - 1 ) )
then  ( ( (((p1 `1) / (p1 `2)) * (p1 `2)) / (p1 `2) >= (p1 `2) / (p1 `2) & (- (p1 `2)) / (p1 `2) >= (((p1 `1) / (p1 `2)) * (p1 `2)) / (p1 `2) ) or ( (((p1 `1) / (p1 `2)) * (p1 `2)) / (p1 `2) <= (p1 `2) / (p1 `2) & (((p1 `1) / (p1 `2)) * (p1 `2)) / (p1 `2) >= (- (p1 `2)) / (p1 `2) ) ) by A96, XREAL_1:73;
then  ( ( (p1 `1) / (p1 `2) >= (p1 `2) / (p1 `2) & (- (p1 `2)) / (p1 `2) >= (p1 `1) / (p1 `2) ) or ( (p1 `1) / (p1 `2) <= (p1 `2) / (p1 `2) & (p1 `1) / (p1 `2) >= (- (p1 `2)) / (p1 `2) ) ) by A100, XCMPLX_1:89;
hence  ( ( (p1 `1) / (p1 `2) <= 1 & - 1 <= (p1 `1) / (p1 `2) ) or ( (p1 `1) / (p1 `2) >= 1 & (p1 `1) / (p1 `2) <= - 1 ) ) by A100, XCMPLX_1:60, XCMPLX_1:197; ::_thesis: verum
end;
end;
end;
 (p2 `2) ^2 > 0 by A94, SQUARE_1:12;
then  1 / (1 + (((p2 `2) / (p2 `1)) ^2)) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A95, XCMPLX_1:60;
then  (1 / (1 + (((p2 `2) / (p2 `1)) ^2))) * (1 + (((p1 `1) / (p1 `2)) ^2)) = ((p1 `2) ^2) / ((p2 `2) ^2) by A76, XCMPLX_1:87;
then  (((p1 `2) ^2) / ((p1 `2) ^2)) / ((p2 `2) ^2) = (((p1 `1) ^2) / ((p2 `1) ^2)) / ((p1 `2) ^2) by A88, XCMPLX_1:48;
then  1 / ((p2 `2) ^2) = (((p1 `1) ^2) / ((p2 `1) ^2)) / ((p1 `2) ^2) by A93, XCMPLX_1:60;
then  (1 / ((p2 `2) ^2)) * ((p2 `1) ^2) = (((p2 `1) ^2) * (((p1 `1) ^2) / ((p2 `1) ^2))) / ((p1 `2) ^2) by XCMPLX_1:74;
then  (1 / ((p2 `2) ^2)) * ((p2 `1) ^2) = ((p1 `1) ^2) / ((p1 `2) ^2) by A84, XCMPLX_1:87;
then  ((p2 `1) ^2) / ((p2 `2) ^2) = ((p1 `1) ^2) / ((p1 `2) ^2) by XCMPLX_1:99;
then  ((p2 `1) / (p2 `2)) ^2 = ((p1 `1) ^2) / ((p1 `2) ^2) by XCMPLX_1:76;
then A101: ((p2 `1) / (p2 `2)) ^2 = ((p1 `1) / (p1 `2)) ^2 by XCMPLX_1:76;
then A102: ( (p2 `1) / (p2 `2) = (p1 `1) / (p1 `2) or (p2 `1) / (p2 `2) = - ((p1 `1) / (p1 `2)) ) by SQUARE_1:40;
A103: now__::_thesis:_(_(_p2_`1_=_((p1_`1)_/_(p1_`2))_*_(p2_`2)_&_(_1_<=_(p1_`1)_/_(p1_`2)_or_-_1_>=_(p1_`1)_/_(p1_`2)_)_)_or_(_p2_`1_=_(-_((p1_`1)_/_(p1_`2)))_*_(p2_`2)_&_(_1_<=_(p1_`1)_/_(p1_`2)_or_-_1_>=_(p1_`1)_/_(p1_`2)_)_)_)
percases ( p2 `1 = ((p1 `1) / (p1 `2)) * (p2 `2) or p2 `1 = (- ((p1 `1) / (p1 `2))) * (p2 `2) ) by A94, A102, XCMPLX_1:87;
caseA104: p2 `1 = ((p1 `1) / (p1 `2)) * (p2 `2) ; ::_thesis: ( 1 <= (p1 `1) / (p1 `2) or - 1 >= (p1 `1) / (p1 `2) )
now__::_thesis:_(_(_p2_`2_>_0_&_(_(_1_<=_(p1_`1)_/_(p1_`2)_&_-_((p1_`1)_/_(p1_`2))_<=_1_)_or_(_1_>=_(p1_`1)_/_(p1_`2)_&_1_<=_-_((p1_`1)_/_(p1_`2))_)_)_)_or_(_p2_`2_<_0_&_(_(_1_<=_(p1_`1)_/_(p1_`2)_&_-_((p1_`1)_/_(p1_`2))_<=_1_)_or_(_1_>=_(p1_`1)_/_(p1_`2)_&_1_<=_-_((p1_`1)_/_(p1_`2))_)_)_)_)
percases ( p2 `2 > 0 or p2 `2 < 0 ) by A94;
case p2 `2 > 0 ; ::_thesis: ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & 1 <= - ((p1 `1) / (p1 `2)) ) )
then  ( ( (p2 `2) / (p2 `2) <= (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) & (- (((p1 `1) / (p1 `2)) * (p2 `2))) / (p2 `2) <= (p2 `2) / (p2 `2) ) or ( (p2 `2) / (p2 `2) >= (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) & (p2 `2) / (p2 `2) <= (- (((p1 `1) / (p1 `2)) * (p2 `2))) / (p2 `2) ) ) by A82, A104, XREAL_1:72;
then A105: ( ( 1 <= (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) & (- (((p1 `1) / (p1 `2)) * (p2 `2))) / (p2 `2) <= 1 ) or ( 1 >= (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) & 1 <= (- (((p1 `1) / (p1 `2)) * (p2 `2))) / (p2 `2) ) ) by A94, XCMPLX_1:60;
 (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) = (p1 `1) / (p1 `2) by A94, XCMPLX_1:89;
hence  ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & 1 <= - ((p1 `1) / (p1 `2)) ) ) by A105, XCMPLX_1:187; ::_thesis: verum
end;
case p2 `2 < 0 ; ::_thesis: ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & 1 <= - ((p1 `1) / (p1 `2)) ) )
then  ( ( (p2 `2) / (p2 `2) >= (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) & (- (((p1 `1) / (p1 `2)) * (p2 `2))) / (p2 `2) >= (p2 `2) / (p2 `2) ) or ( (p2 `2) / (p2 `2) <= (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) & (p2 `2) / (p2 `2) >= (- (((p1 `1) / (p1 `2)) * (p2 `2))) / (p2 `2) ) ) by A82, A104, XREAL_1:73;
then A106: ( ( 1 >= (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) & (- (((p1 `1) / (p1 `2)) * (p2 `2))) / (p2 `2) >= 1 ) or ( 1 <= (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) & 1 >= (- (((p1 `1) / (p1 `2)) * (p2 `2))) / (p2 `2) ) ) by A94, XCMPLX_1:60;
 (((p1 `1) / (p1 `2)) * (p2 `2)) / (p2 `2) = (p1 `1) / (p1 `2) by A94, XCMPLX_1:89;
hence  ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & 1 <= - ((p1 `1) / (p1 `2)) ) ) by A106, XCMPLX_1:187; ::_thesis: verum
end;
end;
end;
then  ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & - 1 >= - (- ((p1 `1) / (p1 `2))) ) ) by XREAL_1:24;
hence  ( 1 <= (p1 `1) / (p1 `2) or - 1 >= (p1 `1) / (p1 `2) ) ; ::_thesis: verum
end;
caseA107: p2 `1 = (- ((p1 `1) / (p1 `2))) * (p2 `2) ; ::_thesis: ( 1 <= (p1 `1) / (p1 `2) or - 1 >= (p1 `1) / (p1 `2) )
now__::_thesis:_(_(_p2_`2_>_0_&_(_(_1_<=_(p1_`1)_/_(p1_`2)_&_-_((p1_`1)_/_(p1_`2))_<=_1_)_or_(_1_>=_(p1_`1)_/_(p1_`2)_&_1_<=_-_((p1_`1)_/_(p1_`2))_)_)_)_or_(_p2_`2_<_0_&_(_(_1_<=_(p1_`1)_/_(p1_`2)_&_-_((p1_`1)_/_(p1_`2))_<=_1_)_or_(_1_>=_(p1_`1)_/_(p1_`2)_&_1_<=_-_((p1_`1)_/_(p1_`2))_)_)_)_)
percases ( p2 `2 > 0 or p2 `2 < 0 ) by A94;
case p2 `2 > 0 ; ::_thesis: ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & 1 <= - ((p1 `1) / (p1 `2)) ) )
then  ( ( (p2 `2) / (p2 `2) <= ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) & (- ((- ((p1 `1) / (p1 `2))) * (p2 `2))) / (p2 `2) <= (p2 `2) / (p2 `2) ) or ( (p2 `2) / (p2 `2) >= ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) & (p2 `2) / (p2 `2) <= (- ((- ((p1 `1) / (p1 `2))) * (p2 `2))) / (p2 `2) ) ) by A82, A107, XREAL_1:72;
then  ( ( 1 <= ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) & (- ((- ((p1 `1) / (p1 `2))) * (p2 `2))) / (p2 `2) <= 1 ) or ( 1 >= ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) & 1 <= (- ((- ((p1 `1) / (p1 `2))) * (p2 `2))) / (p2 `2) ) ) by A94, XCMPLX_1:60;
then A108: ( ( 1 <= - ((p1 `1) / (p1 `2)) & - (((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2)) <= 1 ) or ( 1 >= - ((p1 `1) / (p1 `2)) & 1 <= - (((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2)) ) ) by A94, XCMPLX_1:89, XCMPLX_1:187;
 ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) = - ((p1 `1) / (p1 `2)) by A94, XCMPLX_1:89;
hence  ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & 1 <= - ((p1 `1) / (p1 `2)) ) ) by A108; ::_thesis: verum
end;
case p2 `2 < 0 ; ::_thesis: ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & 1 <= - ((p1 `1) / (p1 `2)) ) )
then  ( ( (p2 `2) / (p2 `2) >= ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) & (- ((- ((p1 `1) / (p1 `2))) * (p2 `2))) / (p2 `2) >= (p2 `2) / (p2 `2) ) or ( (p2 `2) / (p2 `2) <= ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) & (p2 `2) / (p2 `2) >= (- ((- ((p1 `1) / (p1 `2))) * (p2 `2))) / (p2 `2) ) ) by A82, A107, XREAL_1:73;
then  ( ( 1 >= ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) & (- ((- ((p1 `1) / (p1 `2))) * (p2 `2))) / (p2 `2) >= 1 ) or ( 1 <= ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) & 1 >= (- ((- ((p1 `1) / (p1 `2))) * (p2 `2))) / (p2 `2) ) ) by A94, XCMPLX_1:60;
then A109: ( ( 1 >= - ((p1 `1) / (p1 `2)) & - (((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2)) >= 1 ) or ( 1 <= - ((p1 `1) / (p1 `2)) & 1 >= - (((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2)) ) ) by A94, XCMPLX_1:89, XCMPLX_1:187;
 ((- ((p1 `1) / (p1 `2))) * (p2 `2)) / (p2 `2) = - ((p1 `1) / (p1 `2)) by A94, XCMPLX_1:89;
hence  ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & 1 <= - ((p1 `1) / (p1 `2)) ) ) by A109; ::_thesis: verum
end;
end;
end;
then  ( ( 1 <= (p1 `1) / (p1 `2) & - ((p1 `1) / (p1 `2)) <= 1 ) or ( 1 >= (p1 `1) / (p1 `2) & - 1 >= - (- ((p1 `1) / (p1 `2))) ) ) by XREAL_1:24;
hence  ( 1 <= (p1 `1) / (p1 `2) or - 1 >= (p1 `1) / (p1 `2) ) ; ::_thesis: verum
end;
end;
end;
A110: now__::_thesis:_(_(_(p1_`1)_/_(p1_`2)_=_1_&_((p2_`2)_/_(p2_`1))_^2_=_((p1_`1)_/_(p1_`2))_^2_)_or_(_(p1_`1)_/_(p1_`2)_=_-_1_&_((p2_`2)_/_(p2_`1))_^2_=_((p1_`1)_/_(p1_`2))_^2_)_)
percases ( (p1 `1) / (p1 `2) = 1 or (p1 `1) / (p1 `2) = - 1 ) by A103, A97, XXREAL_0:1;
case (p1 `1) / (p1 `2) = 1 ; ::_thesis: ((p2 `2) / (p2 `1)) ^2 = ((p1 `1) / (p1 `2)) ^2
then  ((p2 `1) ^2) / ((p2 `2) ^2) = 1 by A101, XCMPLX_1:76;
then A111: (p2 `1) ^2 = (p2 `2) ^2 by XCMPLX_1:58;
 ((p2 `2) / (p2 `1)) ^2 = ((p2 `2) ^2) / ((p2 `1) ^2) by XCMPLX_1:76;
hence  ((p2 `2) / (p2 `1)) ^2 = ((p1 `1) / (p1 `2)) ^2 by A101, A111, XCMPLX_1:76; ::_thesis: verum
end;
case (p1 `1) / (p1 `2) = - 1 ; ::_thesis: ((p2 `2) / (p2 `1)) ^2 = ((p1 `1) / (p1 `2)) ^2
then  ((p2 `1) ^2) / ((p2 `2) ^2) = 1 by A101, XCMPLX_1:76;
then A112: (p2 `1) ^2 = (p2 `2) ^2 by XCMPLX_1:58;
 ((p2 `2) / (p2 `1)) ^2 = ((p2 `2) ^2) / ((p2 `1) ^2) by XCMPLX_1:76;
hence  ((p2 `2) / (p2 `1)) ^2 = ((p1 `1) / (p1 `2)) ^2 by A101, A112, XCMPLX_1:76; ::_thesis: verum
end;
end;
end;
then  p2 `1 = ((p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A87, A91, XCMPLX_1:87;
then A113: p2 `1 = p1 `1 by A77, XCMPLX_1:87;
 p2 `2 = ((p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A89, A91, A110, XCMPLX_1:87;
then  p2 `2 = p1 `2 by A77, XCMPLX_1:87;
then  p2 = |[(p1 `1),(p1 `2)]| by A113, EUCLID:53;
hence  x1 = x2 by EUCLID:53; ::_thesis: verum
end;
end;
end;
hence  x1 = x2 ; ::_thesis: verum
end;
caseA114: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: x1 = x2
then  p2 `2 <> 0 ;
then A115: (p2 `2) ^2 > 0 by SQUARE_1:12;
A116: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A117: 1 + (((p2 `1) / (p2 `2)) ^2) > 0 by Lm1;
A118: Sq_Circ . p2 = |[((p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| by A114, Def1;
then A119: (p2 `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = (p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A3, A78, A75, EUCLID:52;
then  ((p2 `1) ^2) / ((sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ^2) = ((p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))) ^2 by XCMPLX_1:76;
then  ((p2 `1) ^2) / ((sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ^2) = ((p1 `1) ^2) / ((sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ^2) by XCMPLX_1:76;
then  ((p2 `1) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2)) = ((p1 `1) ^2) / ((sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ^2) by A117, SQUARE_1:def_2;
then A120: ((p2 `1) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2)) = ((p1 `1) ^2) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A76, SQUARE_1:def_2;
A121: (p2 `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = (p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A3, A78, A74, A118, EUCLID:52;
then  ((p2 `2) ^2) / ((sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ^2) = ((p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))) ^2 by XCMPLX_1:76;
then  ((p2 `2) ^2) / ((sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ^2) = ((p1 `2) ^2) / ((sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ^2) by XCMPLX_1:76;
then  ((p2 `2) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2)) = ((p1 `2) ^2) / ((sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ^2) by A117, SQUARE_1:def_2;
then  ((p2 `2) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2)) = ((p1 `2) ^2) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A76, SQUARE_1:def_2;
then  (((p2 `2) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2))) / ((p2 `2) ^2) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by XCMPLX_1:48;
then  (((p2 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p2 `1) / (p2 `2)) ^2)) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by XCMPLX_1:48;
then  1 / (1 + (((p2 `1) / (p2 `2)) ^2)) = (((p1 `2) ^2) / ((p2 `2) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A115, XCMPLX_1:60;
then A122: (1 / (1 + (((p2 `1) / (p2 `2)) ^2))) * (1 + (((p1 `1) / (p1 `2)) ^2)) = ((p1 `2) ^2) / ((p2 `2) ^2) by A76, XCMPLX_1:87;
 p1 `2 <> 0 by A73;
then A123: (p1 `2) ^2 > 0 by SQUARE_1:12;
now__::_thesis:_(_(_p2_`1_=_0_&_x1_=_x2_)_or_(_p2_`1_<>_0_&_x1_=_x2_)_)
percases ( p2 `1 = 0 or p2 `1 <> 0 ) ;
caseA124: p2 `1 = 0 ; ::_thesis: x1 = x2
then  (p1 `1) ^2 = 0 by A76, A120, XCMPLX_1:50;
then A125: p1 `1 = 0 by XCMPLX_1:6;
then  p2 = |[0,(p1 `2)]| by A3, A78, A118, A124, EUCLID:53, SQUARE_1:18;
hence  x1 = x2 by A125, EUCLID:53; ::_thesis: verum
end;
case p2 `1 <> 0 ; ::_thesis: x1 = x2
then A126: (p2 `1) ^2 > 0 by SQUARE_1:12;
 (((p2 `1) ^2) / (1 + (((p2 `1) / (p2 `2)) ^2))) / ((p2 `1) ^2) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A120, XCMPLX_1:48;
then  (((p2 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p2 `1) / (p2 `2)) ^2)) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by XCMPLX_1:48;
then  1 / (1 + (((p2 `1) / (p2 `2)) ^2)) = (((p1 `1) ^2) / ((p2 `1) ^2)) / (1 + (((p1 `1) / (p1 `2)) ^2)) by A126, XCMPLX_1:60;
then  (1 / (1 + (((p2 `1) / (p2 `2)) ^2))) * (1 + (((p1 `1) / (p1 `2)) ^2)) = ((p1 `1) ^2) / ((p2 `1) ^2) by A76, XCMPLX_1:87;
then  (((p1 `2) ^2) / ((p1 `2) ^2)) / ((p2 `2) ^2) = (((p1 `1) ^2) / ((p2 `1) ^2)) / ((p1 `2) ^2) by A122, XCMPLX_1:48;
then  1 / ((p2 `2) ^2) = (((p1 `1) ^2) / ((p2 `1) ^2)) / ((p1 `2) ^2) by A123, XCMPLX_1:60;
then  (1 / ((p2 `2) ^2)) * ((p2 `1) ^2) = (((p2 `1) ^2) * (((p1 `1) ^2) / ((p2 `1) ^2))) / ((p1 `2) ^2) by XCMPLX_1:74;
then  (1 / ((p2 `2) ^2)) * ((p2 `1) ^2) = ((p1 `1) ^2) / ((p1 `2) ^2) by A126, XCMPLX_1:87;
then  ((p2 `1) ^2) / ((p2 `2) ^2) = ((p1 `1) ^2) / ((p1 `2) ^2) by XCMPLX_1:99;
then  ((p2 `1) / (p2 `2)) ^2 = ((p1 `1) ^2) / ((p1 `2) ^2) by XCMPLX_1:76;
then A127: 1 + (((p2 `1) / (p2 `2)) ^2) = 1 + (((p1 `1) / (p1 `2)) ^2) by XCMPLX_1:76;
then  p2 `1 = ((p1 `1) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A119, A116, XCMPLX_1:87;
then A128: p2 `1 = p1 `1 by A77, XCMPLX_1:87;
 p2 `2 = ((p1 `2) / (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A121, A116, A127, XCMPLX_1:87;
then  p2 `2 = p1 `2 by A77, XCMPLX_1:87;
then  p2 = |[(p1 `1),(p1 `2)]| by A128, EUCLID:53;
hence  x1 = x2 by EUCLID:53; ::_thesis: verum
end;
end;
end;
hence  x1 = x2 ; ::_thesis: verum
end;
end;
end;
hence  x1 = x2 ; ::_thesis: verum
end;
end;
end;

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
proof
let Kb, Cb be Subset of (TOP-REAL 2); ::_thesis: ( 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  }  implies Sq_Circ .: Kb = Cb )
assume A1: ( 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  }  ) ; ::_thesis: Sq_Circ .: Kb = Cb
thus  Sq_Circ .: Kb c= Cb :: according to XBOOLE_0:def_10 ::_thesis: Cb c= Sq_Circ .: Kb
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Sq_Circ .: Kb or y in Cb )
assume  y in Sq_Circ .: Kb ; ::_thesis: y in Cb
then consider x being set  such that
 x in dom Sq_Circ and
A2: x in Kb and
A3: y = Sq_Circ . x by FUNCT_1:def_6;
consider q being Point of (TOP-REAL 2) such that
A4: q = x and
A5: ( ( - 1 = q `1 & - 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 ) ) by A1, A2;
now__::_thesis:_(_(_q_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_q_<>_0._(TOP-REAL_2)_&_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_or_(_q_<>_0._(TOP-REAL_2)_&_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_)
percases ( q = 0. (TOP-REAL 2) or ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) or ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ;
case q = 0. (TOP-REAL 2) ; ::_thesis: contradiction
hence  contradiction by A5, JGRAPH_2:3; ::_thesis: verum
end;
caseA6: ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
A7: ( |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2))) & |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))) ) by EUCLID:52;
A8: 1 + ((q `2) ^2) > 0 by Lm1;
A9: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by A6, Def1;
now__::_thesis:_(_(_-_1_=_q_`1_&_-_1_<=_q_`2_&_q_`2_<=_1_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_or_(_q_`1_=_1_&_-_1_<=_q_`2_&_q_`2_<=_1_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_or_(_-_1_=_q_`2_&_-_1_<=_q_`1_&_q_`1_<=_1_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_or_(_1_=_q_`2_&_-_1_<=_q_`1_&_q_`1_<=_1_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_)
percases ( ( - 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 ) ) by A5;
case ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
then |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| ^2 = (((- 1) / (sqrt (1 + (((q `2) / (- 1)) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `2) / (- 1)) ^2)))) ^2) by A7, JGRAPH_1:29
.= (((- 1) ^2) / ((sqrt (1 + (((q `2) / (- 1)) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `2) / (- 1)) ^2)))) ^2) by XCMPLX_1:76
.= (1 / ((sqrt (1 + ((- (q `2)) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + ((- (q `2)) ^2))) ^2)) by XCMPLX_1:76
.= (1 / (1 + ((q `2) ^2))) + (((q `2) ^2) / ((sqrt (1 + ((q `2) ^2))) ^2)) by A8, SQUARE_1:def_2
.= (1 / (1 + ((q `2) ^2))) + (((q `2) ^2) / (1 + ((q `2) ^2))) by A8, SQUARE_1:def_2
.= (1 + ((q `2) ^2)) / (1 + ((q `2) ^2)) by XCMPLX_1:62
.= 1 by A8, XCMPLX_1:60 ;
then  |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| = 1 by SQUARE_1:18, SQUARE_1:22;
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) by A3, A4, A9; ::_thesis: verum
end;
case ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
then |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| ^2 = ((1 / (sqrt (1 + (((q `2) / 1) ^2)))) ^2) + (((q `2) / (sqrt (1 + (((q `2) / 1) ^2)))) ^2) by A7, JGRAPH_1:29
.= ((1 ^2) / ((sqrt (1 + (((q `2) / 1) ^2))) ^2)) + (((q `2) / (sqrt (1 + (((q `2) / 1) ^2)))) ^2) by XCMPLX_1:76
.= (1 / ((sqrt (1 + (((q `2) / 1) ^2))) ^2)) + (((q `2) ^2) / ((sqrt (1 + (((q `2) / 1) ^2))) ^2)) by XCMPLX_1:76
.= (1 / (1 + ((q `2) ^2))) + (((q `2) ^2) / ((sqrt (1 + ((q `2) ^2))) ^2)) by A8, SQUARE_1:def_2
.= (1 / (1 + ((q `2) ^2))) + (((q `2) ^2) / (1 + ((q `2) ^2))) by A8, SQUARE_1:def_2
.= (1 + ((q `2) ^2)) / (1 + ((q `2) ^2)) by XCMPLX_1:62
.= 1 by A8, XCMPLX_1:60 ;
then  |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| = 1 by SQUARE_1:18, SQUARE_1:22;
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) by A3, A4, A9; ::_thesis: verum
end;
caseA10: ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
then  ( ( - 1 <= q `1 & q `1 >= 1 ) or ( - 1 >= q `1 & 1 >= q `1 ) ) by A6, XREAL_1:24;
then A11: ( q `1 = 1 or q `1 = - 1 ) by A10, XXREAL_0:1;
|.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((- 1) / (q `1)) ^2)))) ^2) + (((- 1) / (sqrt (1 + (((- 1) / (q `1)) ^2)))) ^2) by A7, A10, JGRAPH_1:29
.= (((q `1) / (sqrt (1 + (((- 1) / (q `1)) ^2)))) ^2) + (((- 1) ^2) / ((sqrt (1 + (((- 1) / (q `1)) ^2))) ^2)) by XCMPLX_1:76
.= (((q `1) ^2) / ((sqrt (1 + (((- 1) / (q `1)) ^2))) ^2)) + (1 / ((sqrt (1 + (((- 1) / (q `1)) ^2))) ^2)) by XCMPLX_1:76
.= (1 / 2) + (1 / ((sqrt 2) ^2)) by A11, SQUARE_1:def_2
.= (1 / 2) + (1 / 2) by SQUARE_1:def_2
.= 1 ;
then  |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| = 1 by SQUARE_1:18, SQUARE_1:22;
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) by A3, A4, A9; ::_thesis: verum
end;
caseA12: ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
then  ( ( 1 <= q `1 & q `1 >= - 1 ) or ( 1 >= q `1 & - 1 >= q `1 ) ) by A6, XREAL_1:25;
then A13: ( q `1 = 1 or q `1 = - 1 ) by A12, XXREAL_0:1;
|.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + ((1 / (q `1)) ^2)))) ^2) + ((1 / (sqrt (1 + ((1 / (q `1)) ^2)))) ^2) by A7, A12, JGRAPH_1:29
.= (((q `1) / (sqrt (1 + ((1 / (q `1)) ^2)))) ^2) + ((1 ^2) / ((sqrt (1 + ((1 / (q `1)) ^2))) ^2)) by XCMPLX_1:76
.= (1 / ((sqrt (1 + (1 / 1))) ^2)) + (1 / ((sqrt (1 + (1 / 1))) ^2)) by A13, XCMPLX_1:76
.= (1 / 2) + (1 / ((sqrt 2) ^2)) by SQUARE_1:def_2
.= (1 / 2) + (1 / 2) by SQUARE_1:def_2
.= 1 ;
then  |.|[((q `1) / (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) / (sqrt (1 + (((q `2) / (q `1)) ^2))))]|.| = 1 by SQUARE_1:18, SQUARE_1:22;
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) by A3, A4, A9; ::_thesis: verum
end;
end;
end;
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) ; ::_thesis: verum
end;
caseA14: ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
A15: ( |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `1 = (q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2))) & |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `2 = (q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))) ) by EUCLID:52;
A16: 1 + ((q `1) ^2) > 0 by Lm1;
A17: Sq_Circ . q = |[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by A14, Def1;
now__::_thesis:_(_(_-_1_=_q_`2_&_-_1_<=_q_`1_&_q_`1_<=_1_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_or_(_q_`2_=_1_&_-_1_<=_q_`1_&_q_`1_<=_1_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_or_(_-_1_=_q_`1_&_-_1_<=_q_`2_&_q_`2_<=_1_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_or_(_1_=_q_`1_&_-_1_<=_q_`2_&_q_`2_<=_1_&_ex_p2_being_Point_of_(TOP-REAL_2)_st_
(_p2_=_y_&_|.p2.|_=_1_)_)_)
percases ( ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( q `2 = 1 & - 1 <= q `1 & q `1 <= 1 ) or ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) ) by A5;
case ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
then |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| ^2 = (((q `1) / (sqrt (1 + (((q `1) / (- 1)) ^2)))) ^2) + (((- 1) / (sqrt (1 + (((q `1) / (- 1)) ^2)))) ^2) by A15, JGRAPH_1:29
.= (((- 1) ^2) / ((sqrt (1 + (((q `1) / (- 1)) ^2))) ^2)) + (((q `1) / (sqrt (1 + (((q `1) / (- 1)) ^2)))) ^2) by XCMPLX_1:76
.= (1 / ((sqrt (1 + ((- (q `1)) ^2))) ^2)) + (((q `1) ^2) / ((sqrt (1 + ((- (q `1)) ^2))) ^2)) by XCMPLX_1:76
.= (1 / (1 + ((q `1) ^2))) + (((q `1) ^2) / ((sqrt (1 + ((q `1) ^2))) ^2)) by A16, SQUARE_1:def_2
.= (1 / (1 + ((q `1) ^2))) + (((q `1) ^2) / (1 + ((q `1) ^2))) by A16, SQUARE_1:def_2
.= (1 + ((q `1) ^2)) / (1 + ((q `1) ^2)) by XCMPLX_1:62
.= 1 by A16, XCMPLX_1:60 ;
then  |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| = 1 by SQUARE_1:18, SQUARE_1:22;
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) by A3, A4, A17; ::_thesis: verum
end;
case ( q `2 = 1 & - 1 <= q `1 & q `1 <= 1 ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
then |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| ^2 = ((1 / (sqrt (1 + (((q `1) / 1) ^2)))) ^2) + (((q `1) / (sqrt (1 + (((q `1) / 1) ^2)))) ^2) by A15, JGRAPH_1:29
.= ((1 ^2) / ((sqrt (1 + (((q `1) / 1) ^2))) ^2)) + (((q `1) / (sqrt (1 + (((q `1) / 1) ^2)))) ^2) by XCMPLX_1:76
.= (1 / ((sqrt (1 + (((q `1) / 1) ^2))) ^2)) + (((q `1) ^2) / ((sqrt (1 + (((q `1) / 1) ^2))) ^2)) by XCMPLX_1:76
.= (1 / (1 + ((q `1) ^2))) + (((q `1) ^2) / ((sqrt (1 + ((q `1) ^2))) ^2)) by A16, SQUARE_1:def_2
.= (1 / (1 + ((q `1) ^2))) + (((q `1) ^2) / (1 + ((q `1) ^2))) by A16, SQUARE_1:def_2
.= (1 + ((q `1) ^2)) / (1 + ((q `1) ^2)) by XCMPLX_1:62
.= 1 by A16, XCMPLX_1:60 ;
then  |.|[((q `1) / (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) / (sqrt (1 + (((q `1) / (q `2)) ^2))))]|.| = 1 by SQUARE_1:18, SQUARE_1:22;
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) by A3, A4, A17; ::_thesis: verum
end;
case ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) by A14; ::_thesis: verum
end;
case ( 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 )
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) by A14; ::_thesis: verum
end;
end;
end;
hence  ex p2 being Point of (TOP-REAL 2) st
( p2 = y & |.p2.| = 1 ) ; ::_thesis: verum
end;
end;
end;
hence  y in Cb by A1; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Cb or y in Sq_Circ .: Kb )
assume  y in Cb ; ::_thesis: y in Sq_Circ .: Kb
then consider p2 being Point of (TOP-REAL 2) such that
A18: p2 = y and
A19: |.p2.| = 1 by A1;
set q = p2;
now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_ex_x_being_set_st_
(_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_ex_x_being_set_st_
(_x_in_dom_Sq_Circ_&_x_in_Kb_&_y_=_Sq_Circ_._x_)_)_)
percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ;
case p2 = 0. (TOP-REAL 2) ; ::_thesis: contradiction
hence  contradiction by A19, TOPRNS_1:23; ::_thesis: verum
end;
caseA20: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: ex x being set st
( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x )
A21: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_1:29;
set px = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]|;
A22: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52;
A23: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
then A24: p2 `2 = ((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by XCMPLX_1:89
.= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ;
A25: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52;
then A26: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) = (p2 `2) / (p2 `1) by A22, A23, XCMPLX_1:91;
then A27: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `2 by A25, A23, XCMPLX_1:89;
 ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A20, A23, XREAL_1:64;
then A28: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A22, A25, A23, XREAL_1:64;
A29: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2) > 0 by Lm1;
p2 `1 = ((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A23, XCMPLX_1:89
.= (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52 ;
then  (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) ^2) = 1 by A19, A26, A24, A21, XCMPLX_1:76;
then  (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) = 1 by XCMPLX_1:76;
then  (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ^2)) = 1 by A29, SQUARE_1:def_2;
then 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) = (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) * ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))) by A29, SQUARE_1:def_2
.= ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) ;
then  ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A29, XCMPLX_1:87;
then A30: ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) = 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)) by A29, XCMPLX_1:87
.= 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2)) by XCMPLX_1:76 ;
A31: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_=_0_)
assume that
A32: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 and
A33: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = 0 ; ::_thesis: contradiction
 (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A33, EUCLID:52;
then A34: p2 `2 = 0 by A23, XCMPLX_1:6;
 (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A32, EUCLID:52;
then  p2 `1 = 0 by A23, XCMPLX_1:6;
hence  contradiction by A20, A34, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
then  not |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 by A22, A25, A23, A28, XREAL_1:64;
then  (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) = (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2 by A30, XCMPLX_1:6, XCMPLX_1:87;
then  0 = (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2)) ;
then A35: ( ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) - 1 = 0 or ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) ^2) = 0 ) by XCMPLX_1:6;
now__::_thesis:_(_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_1_&_(_(_-_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_<=_1_)_or_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_<=_1_)_or_(_-_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_<=_1_)_or_(_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_<=_1_)_)_)_or_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_-_1_&_(_(_-_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_<=_1_)_or_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_<=_1_)_or_(_-_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_<=_1_)_or_(_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_<=_1_)_)_)_)
percases ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 1 or |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = - 1 ) by A31, A35, COMPLEX1:1, SQUARE_1:41;
case |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 1 ; ::_thesis: ( ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 ) or ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ) or ( 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ) )
hence  ( ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 ) or ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ) or ( 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ) ) by A22, A25, A23, A28, XREAL_1:64; ::_thesis: verum
end;
case |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = - 1 ; ::_thesis: ( ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 ) or ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ) or ( 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ) )
hence  ( ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= 1 ) or ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ) or ( 1 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 <= 1 ) ) by A22, A23, A28, XREAL_1:64; ::_thesis: verum
end;
end;
end;
then A36: ( dom Sq_Circ = the carrier of (TOP-REAL 2) & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| in Kb ) by A1, FUNCT_2:def_1;
 ( ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A22, A25, A23, A28, XREAL_1:64;
then A37: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))))]| by A31, Def1, JGRAPH_2:3;
 (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `1 by A22, A23, A26, XCMPLX_1:89;
hence  ex x being set st
( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A18, A37, A27, A36, EUCLID:53; ::_thesis: verum
end;
caseA38: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: ex x being set st
( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x )
A39: |.p2.| ^2 = ((p2 `2) ^2) + ((p2 `1) ^2) by JGRAPH_1:29;
set px = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]|;
A40: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A41: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52;
then A42: p2 `1 = (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A40, XCMPLX_1:89;
A43: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52;
then A44: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) = (p2 `1) / (p2 `2) by A41, A40, XCMPLX_1:91;
then A45: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `1 by A41, A40, XCMPLX_1:89;
 ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A38, JGRAPH_2:13;
then  ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A40, XREAL_1:64;
then A46: ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A43, A41, A40, XREAL_1:64;
A47: 1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2) > 0 by Lm1;
 p2 `2 = (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A43, A40, XCMPLX_1:89;
then  (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) ^2) = 1 by A19, A44, A42, A39, XCMPLX_1:76;
then  (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) = 1 by XCMPLX_1:76;
then  (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ^2)) = 1 by A47, SQUARE_1:def_2;
then 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) = (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) * ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))) by A47, SQUARE_1:def_2
.= ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) ;
then  ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A47, XCMPLX_1:87;
then  ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) = 1 * (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)) by A47, XCMPLX_1:87;
then A48: (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) - 1 = ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) / ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) by XCMPLX_1:76;
A49: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_=_0_)
assume that
A50: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 and
 |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = 0 ; ::_thesis: contradiction
 p2 `2 = 0 by A43, A40, A50, XCMPLX_1:6;
hence  contradiction by A38; ::_thesis: verum
end;
then  |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <> 0 by A43, A41, A40, A46, XREAL_1:64;
then  (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) - 1)) * ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) = (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2 by A48, XCMPLX_1:6, XCMPLX_1:87;
then  0 = (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1) * (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2)) ;
then A51: ( ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) - 1 = 0 or ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ^2) + ((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) ^2) = 0 ) by XCMPLX_1:6;
now__::_thesis:_(_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_1_&_(_(_-_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_<=_1_)_or_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_<=_1_)_or_(_-_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_<=_1_)_or_(_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_<=_1_)_)_)_or_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_-_1_&_(_(_-_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_<=_1_)_or_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_<=_1_)_or_(_-_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_<=_1_)_or_(_1_=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_&_-_1_<=_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_&_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_<=_1_)_)_)_)
percases ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 1 or |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = - 1 ) by A49, A51, COMPLEX1:1, SQUARE_1:41;
case |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 1 ; ::_thesis: ( ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 ) or ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ) or ( 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ) )
hence  ( ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 ) or ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ) or ( 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ) ) by A43, A41, A40, A46, XREAL_1:64; ::_thesis: verum
end;
case |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = - 1 ; ::_thesis: ( ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 ) or ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ) or ( 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ) )
hence  ( ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= 1 ) or ( - 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ) or ( 1 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 & - 1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 <= 1 ) ) by A43, A40, A46, XREAL_1:64; ::_thesis: verum
end;
end;
end;
then A52: ( dom Sq_Circ = the carrier of (TOP-REAL 2) & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| in Kb ) by A1, FUNCT_2:def_1;
 ( ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A43, A41, A40, A46, XREAL_1:64;
then A53: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))))]| by A49, Th4, JGRAPH_2:3;
 (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `2 by A43, A40, A44, XCMPLX_1:89;
hence  ex x being set st
( x in dom Sq_Circ & x in Kb & y = Sq_Circ . x ) by A18, A53, A45, A52, EUCLID:53; ::_thesis: verum
end;
end;
end;
hence  y in Sq_Circ .: Kb by FUNCT_1:def_6; ::_thesis: verum
end;

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
proof
set X = (TOP-REAL 2) | R^2-unit_square;
set b = 1;
set a = 0 ;
set v = |[1,0]|;
let P, Kb be Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | Kb),((TOP-REAL 2) | P); ::_thesis: ( 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 implies P is being_simple_closed_curve )
assume A1: ( 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 ) ; ::_thesis: P is being_simple_closed_curve
 ( |[1,0]| `1 = 1 & |[1,0]| `2 = 0 ) by EUCLID:52;
then A2: |[1,0]| in  {  q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) )  }  ;
then reconsider Kbb = Kb as non empty Subset of (TOP-REAL 2) by A1;
set A = 2 / (1 - 0);
set B = 1 - ((2 * 1) / (1 - 0));
set C = 2 / (1 - 0);
set D = 1 - ((2 * 1) / (1 - 0));
reconsider Kbd = Kbb as non empty Subset of (TOP-REAL 2) ;
defpred S1[ set , set ] means  for t being Point of (TOP-REAL 2) st t = $1 holds
$2 = |[(((2 / (1 - 0)) * (t `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (t `2)) + (1 - ((2 * 1) / (1 - 0))))]|;
A3: for x being set st x in the carrier of (TOP-REAL 2) holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL 2) implies ex y being set st S1[x,y] )
assume  x in the carrier of (TOP-REAL 2) ; ::_thesis: ex y being set st S1[x,y]
then reconsider t2 = x as Point of (TOP-REAL 2) ;
reconsider y2 = |[(((2 / (1 - 0)) * (t2 `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (t2 `2)) + (1 - ((2 * 1) / (1 - 0))))]| as set ;
 for t being Point of (TOP-REAL 2) st t = x holds
y2 = |[(((2 / (1 - 0)) * (t `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (t `2)) + (1 - ((2 * 1) / (1 - 0))))]| ;
hence  ex y being set st S1[x,y] ; ::_thesis: verum
end;
 ex ff being Function st
( dom ff = the carrier of (TOP-REAL 2) & ( for x being set st x in the carrier of (TOP-REAL 2) holds
S1[x,ff . x] ) ) from CLASSES1:sch_1(A3);
then consider ff being Function such that
A4: dom ff = the carrier of (TOP-REAL 2) and
A5: for x being set st x in the carrier of (TOP-REAL 2) holds
for t being Point of (TOP-REAL 2) st t = x holds
ff . x = |[(((2 / (1 - 0)) * (t `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (t `2)) + (1 - ((2 * 1) / (1 - 0))))]| ;
A6: for t being Point of (TOP-REAL 2) holds ff . t = |[(((2 / (1 - 0)) * (t `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (t `2)) + (1 - ((2 * 1) / (1 - 0))))]| by A5;
 for x being set st x in the carrier of (TOP-REAL 2) holds
ff . x in the carrier of (TOP-REAL 2)
proof
let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL 2) implies ff . x in the carrier of (TOP-REAL 2) )
assume  x in the carrier of (TOP-REAL 2) ; ::_thesis: ff . x in the carrier of (TOP-REAL 2)
then reconsider t = x as Point of (TOP-REAL 2) ;
 ff . t = |[(((2 / (1 - 0)) * (t `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (t `2)) + (1 - ((2 * 1) / (1 - 0))))]| by A5;
hence  ff . x in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider ff = ff as Function of (TOP-REAL 2),(TOP-REAL 2) by A4, FUNCT_2:3;
reconsider f11 = ff | R^2-unit_square as Function of ((TOP-REAL 2) | R^2-unit_square),(TOP-REAL 2) by PRE_TOPC:9;
A7: f11 is continuous by A6, JGRAPH_2:43, TOPMETR:7;
 ff is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom ff or not x2 in dom ff or not ff . x1 = ff . x2 or x1 = x2 )
assume that
A8: ( x1 in dom ff & x2 in dom ff ) and
A9: ff . x1 = ff . x2 ; ::_thesis: x1 = x2
reconsider p1 = x1, p2 = x2 as Point of (TOP-REAL 2) by A8;
A10: ( ff . x1 = |[(((2 / (1 - 0)) * (p1 `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (p1 `2)) + (1 - ((2 * 1) / (1 - 0))))]| & ff . x2 = |[(((2 / (1 - 0)) * (p2 `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (p2 `2)) + (1 - ((2 * 1) / (1 - 0))))]| ) by A5;
then  (((2 / (1 - 0)) * (p1 `1)) + (1 - ((2 * 1) / (1 - 0)))) - (1 - ((2 * 1) / (1 - 0))) = (((2 / (1 - 0)) * (p2 `1)) + (1 - ((2 * 1) / (1 - 0)))) - (1 - ((2 * 1) / (1 - 0))) by A9, SPPOL_2:1;
then  ((2 / (1 - 0)) * (p1 `1)) / (2 / (1 - 0)) = p2 `1 by XCMPLX_1:89;
then A11: p1 `1 = p2 `1 by XCMPLX_1:89;
 (((2 / (1 - 0)) * (p1 `2)) + (1 - ((2 * 1) / (1 - 0)))) - (1 - ((2 * 1) / (1 - 0))) = (((2 / (1 - 0)) * (p2 `2)) + (1 - ((2 * 1) / (1 - 0)))) - (1 - ((2 * 1) / (1 - 0))) by A9, A10, SPPOL_2:1;
then  ((2 / (1 - 0)) * (p1 `2)) / (2 / (1 - 0)) = p2 `2 by XCMPLX_1:89;
hence  x1 = x2 by A11, TOPREAL3:6, XCMPLX_1:89; ::_thesis: verum
end;
then A12: f11 is one-to-one by FUNCT_1:52;
A13: dom f11 = (dom ff) /\ R^2-unit_square by RELAT_1:61
.= R^2-unit_square by A4, XBOOLE_1:28 ;
A14: Kbd c= rng f11
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Kbd or y in rng f11 )
assume A15: y in Kbd ; ::_thesis: y in rng f11
then reconsider py = y as Point of (TOP-REAL 2) ;
set t = |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]|;
A16: ex q being Point of (TOP-REAL 2) st
( py = q & ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) ) by A1, A15;
now__::_thesis:_(_(_-_1_=_py_`1_&_-_1_<=_py_`2_&_py_`2_<=_1_&_(_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_>=_0_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_>=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_=_1_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_>=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_=_0_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_>=_0_)_)_)_or_(_py_`1_=_1_&_-_1_<=_py_`2_&_py_`2_<=_1_&_(_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_>=_0_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_>=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_=_1_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_>=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_=_0_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_>=_0_)_)_)_or_(_-_1_=_py_`2_&_-_1_<=_py_`1_&_py_`1_<=_1_&_(_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_>=_0_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_>=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_=_1_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_>=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_=_0_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_>=_0_)_)_)_or_(_1_=_py_`2_&_-_1_<=_py_`1_&_py_`1_<=_1_&_(_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_>=_0_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_>=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_=_1_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_>=_0_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_=_0_)_or_(_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`1_=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_<=_1_&_|[(((py_`1)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2),(((py_`2)_-_(1_-_((2_*_1)_/_(1_-_0))))_/_2)]|_`2_>=_0_)_)_)_)
percases ( ( - 1 = py `1 & - 1 <= py `2 & py `2 <= 1 ) or ( py `1 = 1 & - 1 <= py `2 & py `2 <= 1 ) or ( - 1 = py `2 & - 1 <= py `1 & py `1 <= 1 ) or ( 1 = py `2 & - 1 <= py `1 & py `1 <= 1 ) ) by A16;
caseA17: ( - 1 = py `1 & - 1 <= py `2 & py `2 <= 1 ) ; ::_thesis: ( ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 1 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) )
then  2 - 1 >= py `2 ;
then  2 >= (py `2) + 1 by XREAL_1:19;
then A18: 2 / 2 >= ((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2 by XREAL_1:72;
 0 - 1 <= py `2 by A17;
then  0 <= (py `2) + 1 by XREAL_1:20;
hence  ( ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 1 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) ) by A17, A18, EUCLID:52; ::_thesis: verum
end;
caseA19: ( py `1 = 1 & - 1 <= py `2 & py `2 <= 1 ) ; ::_thesis: ( ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 1 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) )
then  2 - 1 >= py `2 ;
then  2 >= (py `2) + 1 by XREAL_1:19;
then A20: 2 / 2 >= ((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2 by XREAL_1:72;
 0 - 1 <= py `2 by A19;
then  0 <= (py `2) + 1 by XREAL_1:20;
hence  ( ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 1 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) ) by A19, A20, EUCLID:52; ::_thesis: verum
end;
caseA21: ( - 1 = py `2 & - 1 <= py `1 & py `1 <= 1 ) ; ::_thesis: ( ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 1 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) )
then  2 - 1 >= py `1 ;
then  2 >= (py `1) + 1 by XREAL_1:19;
then A22: 2 / 2 >= ((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2 by XREAL_1:72;
 0 - 1 <= py `1 by A21;
then  0 <= (py `1) + 1 by XREAL_1:20;
hence  ( ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 1 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) ) by A21, A22, EUCLID:52; ::_thesis: verum
end;
caseA23: ( 1 = py `2 & - 1 <= py `1 & py `1 <= 1 ) ; ::_thesis: ( ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 1 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) )
then  2 - 1 >= py `1 ;
then  2 >= (py `1) + 1 by XREAL_1:19;
then A24: 2 / 2 >= ((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2 by XREAL_1:72;
 0 - 1 <= py `1 by A23;
then  0 <= (py `1) + 1 by XREAL_1:20;
hence  ( ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 1 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 >= 0 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = 0 ) or ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 <= 1 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 >= 0 ) ) by A23, A24, EUCLID:52; ::_thesis: verum
end;
end;
end;
then A25: |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| in R^2-unit_square by TOPREAL1:14;
 ( |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1 = ((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2 & |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2 = ((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2 ) by EUCLID:52;
then  py = |[(((2 / (1 - 0)) * (|[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (|[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| `2)) + (1 - ((2 * 1) / (1 - 0))))]| by EUCLID:53;
then py = ff . |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| by A5
.= f11 . |[(((py `1) - (1 - ((2 * 1) / (1 - 0)))) / 2),(((py `2) - (1 - ((2 * 1) / (1 - 0)))) / 2)]| by A25, FUNCT_1:49 ;
hence  y in rng f11 by A13, A25, FUNCT_1:def_3; ::_thesis: verum
end;
 rng f11 c= Kbd
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f11 or y in Kbd )
assume  y in rng f11 ; ::_thesis: y in Kbd
then consider x being set  such that
A26: x in dom f11 and
A27: y = f11 . x by FUNCT_1:def_3;
reconsider t = x as Point of (TOP-REAL 2) by A13, A26;
A28: y = ff . t by A13, A26, A27, FUNCT_1:49
.= |[(((2 / (1 - 0)) * (t `1)) + (1 - ((2 * 1) / (1 - 0)))),(((2 / (1 - 0)) * (t `2)) + (1 - ((2 * 1) / (1 - 0))))]| by A5 ;
then reconsider qy = y as Point of (TOP-REAL 2) ;
A29: ex p being Point of (TOP-REAL 2) st
( t = p & ( ( p `1 = 0 & p `2 <= 1 & p `2 >= 0 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 1 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 0 ) or ( p `1 = 1 & p `2 <= 1 & p `2 >= 0 ) ) ) by A13, A26, TOPREAL1:14;
now__::_thesis:_(_(_-_1_=_qy_`1_&_-_1_<=_qy_`2_&_qy_`2_<=_1_)_or_(_qy_`1_=_1_&_-_1_<=_qy_`2_&_qy_`2_<=_1_)_or_(_-_1_=_qy_`2_&_-_1_<=_qy_`1_&_qy_`1_<=_1_)_or_(_1_=_qy_`2_&_-_1_<=_qy_`1_&_qy_`1_<=_1_)_)
percases ( ( t `1 = 0 & t `2 <= 1 & t `2 >= 0 ) or ( t `1 <= 1 & t `1 >= 0 & t `2 = 1 ) or ( t `1 <= 1 & t `1 >= 0 & t `2 = 0 ) or ( t `1 = 1 & t `2 <= 1 & t `2 >= 0 ) ) by A29;
supposeA30: ( t `1 = 0 & t `2 <= 1 & t `2 >= 0 ) ; ::_thesis: ( ( - 1 = qy `1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( qy `1 = 1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( - 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) or ( 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) )
A31: qy `2 = (2 * (t `2)) - 1 by A28, EUCLID:52;
 2 * 1 >= 2 * (t `2) by A30, XREAL_1:64;
then A32: (1 + 1) - 1 >= ((qy `2) + 1) - 1 by A31, XREAL_1:9;
 0 - 1 <= ((qy `2) + 1) - 1 by A30, A31, XREAL_1:9;
hence  ( ( - 1 = qy `1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( qy `1 = 1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( - 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) or ( 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) ) by A28, A30, A32, EUCLID:52; ::_thesis: verum
end;
supposeA33: ( t `1 <= 1 & t `1 >= 0 & t `2 = 1 ) ; ::_thesis: ( ( - 1 = qy `1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( qy `1 = 1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( - 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) or ( 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) )
A34: qy `1 = (2 * (t `1)) - 1 by A28, EUCLID:52;
 2 * 1 >= 2 * (t `1) by A33, XREAL_1:64;
then A35: (1 + 1) - 1 >= ((qy `1) + 1) - 1 by A34, XREAL_1:9;
 0 - 1 <= ((qy `1) + 1) - 1 by A33, A34, XREAL_1:9;
hence  ( ( - 1 = qy `1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( qy `1 = 1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( - 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) or ( 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) ) by A28, A33, A35, EUCLID:52; ::_thesis: verum
end;
supposeA36: ( t `1 <= 1 & t `1 >= 0 & t `2 = 0 ) ; ::_thesis: ( ( - 1 = qy `1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( qy `1 = 1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( - 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) or ( 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) )
A37: qy `1 = (2 * (t `1)) - 1 by A28, EUCLID:52;
 2 * 1 >= 2 * (t `1) by A36, XREAL_1:64;
then A38: (1 + 1) - 1 >= ((qy `1) + 1) - 1 by A37, XREAL_1:9;
 0 - 1 <= ((qy `1) + 1) - 1 by A36, A37, XREAL_1:9;
hence  ( ( - 1 = qy `1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( qy `1 = 1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( - 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) or ( 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) ) by A28, A36, A38, EUCLID:52; ::_thesis: verum
end;
supposeA39: ( t `1 = 1 & t `2 <= 1 & t `2 >= 0 ) ; ::_thesis: ( ( - 1 = qy `1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( qy `1 = 1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( - 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) or ( 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) )
A40: qy `2 = (2 * (t `2)) - 1 by A28, EUCLID:52;
 2 * 1 >= 2 * (t `2) by A39, XREAL_1:64;
then A41: (1 + 1) - 1 >= ((qy `2) + 1) - 1 by A40, XREAL_1:9;
 0 - 1 <= ((qy `2) + 1) - 1 by A39, A40, XREAL_1:9;
hence  ( ( - 1 = qy `1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( qy `1 = 1 & - 1 <= qy `2 & qy `2 <= 1 ) or ( - 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) or ( 1 = qy `2 & - 1 <= qy `1 & qy `1 <= 1 ) ) by A28, A39, A41, EUCLID:52; ::_thesis: verum
end;
end;
end;
hence  y in Kbd by A1; ::_thesis: verum
end;
then  Kbd = rng f11 by A14, XBOOLE_0:def_10;
then consider f1 being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | Kbd) such that
 f11 = f1 and
A42: f1 is being_homeomorphism by A7, A12, JGRAPH_1:46;
dom f = [#] ((TOP-REAL 2) | Kb) by A1, TOPS_2:def_5
.= Kb by PRE_TOPC:def_5 ;
then  f . |[1,0]| in rng f by A1, A2, FUNCT_1:3;
then reconsider PP = P as non empty Subset of (TOP-REAL 2) ;
reconsider g = f as Function of ((TOP-REAL 2) | Kbb),((TOP-REAL 2) | PP) ;
reconsider g = g as Function of ((TOP-REAL 2) | Kbb),((TOP-REAL 2) | PP) ;
reconsider f22 = f1 as Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | Kbb) ;
reconsider h = g * f22 as Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | PP) ;
 h is being_homeomorphism by A1, A42, TOPS_2:57;
hence  P is being_simple_closed_curve by TOPREAL2:def_1; ::_thesis: verum
end;

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 )
proof
set v = |[1,0]|;
let Kb be Subset of (TOP-REAL 2); ::_thesis: ( 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 ) )  }  implies ( Kb is being_simple_closed_curve & Kb is compact ) )
assume A1: 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 ) )  }  ; ::_thesis: ( Kb is being_simple_closed_curve & Kb is compact )
 ( |[1,0]| `1 = 1 & |[1,0]| `2 = 0 ) by EUCLID:52;
then  |[1,0]| in  {  q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) )  }  ;
then reconsider Kbd = Kb as non empty Subset of (TOP-REAL 2) by A1;
set P = Kb;
 id ((TOP-REAL 2) | Kbd) is being_homeomorphism ;
hence  Kb is being_simple_closed_curve by A1, Th24; ::_thesis: Kb is compact
then consider f being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | Kb) such that
A2: f is being_homeomorphism by TOPREAL2:def_1;
percases ( Kb is empty or not Kb is empty ) ;
supposeA3: Kb is empty ; ::_thesis: Kb is compact
 Kbd <> {} ;
hence  Kb is compact by A3; ::_thesis: verum
end;
suppose not Kb is empty ; ::_thesis: Kb is compact
then reconsider R = Kb as non empty Subset of (TOP-REAL 2) ;
 ( f is continuous & rng f = [#] ((TOP-REAL 2) | Kb) ) by A2, TOPS_2:def_5;
then  (TOP-REAL 2) | R is compact by COMPTS_1:14;
hence  Kb is compact by COMPTS_1:3; ::_thesis: verum
end;
end;
end;

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
proof
defpred S1[ Point of (TOP-REAL 2)] means ( ( - 1 = $1 `1 & - 1 <= $1 `2 & $1 `2 <= 1 ) or ( $1 `1 = 1 & - 1 <= $1 `2 & $1 `2 <= 1 ) or ( - 1 = $1 `2 & - 1 <= $1 `1 & $1 `1 <= 1 ) or ( 1 = $1 `2 & - 1 <= $1 `1 & $1 `1 <= 1 ) );
A1: ( |[1,0]| `1 = 1 & |[1,0]| `2 = 0 ) by EUCLID:52;
A2: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
set v = |[1,0]|;
let Cb be Subset of (TOP-REAL 2); ::_thesis: ( Cb =  {  p where p is Point of (TOP-REAL 2) : |.p.| = 1  }  implies Cb is being_simple_closed_curve )
assume A3: Cb =  {  p where p is Point of (TOP-REAL 2) : |.p.| = 1  }  ; ::_thesis: Cb is being_simple_closed_curve
 ( |[1,0]| `1 = 1 & |[1,0]| `2 = 0 ) by EUCLID:52;
then A4: |[1,0]| in  {  q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) )  }  ;
  {  q where q is Element of (TOP-REAL 2) : S1[q]  }  is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7();
then reconsider Kb =  {  q where q is Point of (TOP-REAL 2) : ( ( - 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 ) )  }  as non empty Subset of (TOP-REAL 2) by A4;
|.|[1,0]|.| = sqrt (((|[1,0]| `1) ^2) + ((|[1,0]| `2) ^2)) by JGRAPH_1:30
.= 1 by A1, SQUARE_1:18 ;
then  |[1,0]| in Cb by A3;
then reconsider Cbb = Cb as non empty Subset of (TOP-REAL 2) ;
A5: the carrier of ((TOP-REAL 2) | Kb) = Kb by PRE_TOPC:8;
A6: dom (Sq_Circ | Kb) = (dom Sq_Circ) /\ Kb by RELAT_1:61
.= the carrier of ((TOP-REAL 2) | Kb) by A5, A2, XBOOLE_1:28 ;
A7: rng (Sq_Circ | Kb) c= (Sq_Circ | Kb) .: the carrier of ((TOP-REAL 2) | Kb)
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng (Sq_Circ | Kb) or u in (Sq_Circ | Kb) .: the carrier of ((TOP-REAL 2) | Kb) )
assume  u in rng (Sq_Circ | Kb) ; ::_thesis: u in (Sq_Circ | Kb) .: the carrier of ((TOP-REAL 2) | Kb)
then  ex z being set st
( z in dom (Sq_Circ | Kb) & u = (Sq_Circ | Kb) . z ) by FUNCT_1:def_3;
hence  u in (Sq_Circ | Kb) .: the carrier of ((TOP-REAL 2) | Kb) by A6, FUNCT_1:def_6; ::_thesis: verum
end;
(Sq_Circ | Kb) .: the carrier of ((TOP-REAL 2) | Kb) = Sq_Circ .: Kb by A5, RELAT_1:129
.= Cb by A3, Th23
.= the carrier of ((TOP-REAL 2) | Cbb) by PRE_TOPC:8 ;
then reconsider f0 = Sq_Circ | Kb as Function of ((TOP-REAL 2) | Kb),((TOP-REAL 2) | Cbb) by A6, A7, FUNCT_2:2;
 rng (Sq_Circ | Kb) c= the carrier of (TOP-REAL 2) ;
then reconsider f00 = f0 as Function of ((TOP-REAL 2) | Kb),(TOP-REAL 2) by A6, FUNCT_2:2;
A8: ( f0 is one-to-one & Kb is compact ) by Th25, FUNCT_1:52;
rng f0 = (Sq_Circ | Kb) .: the carrier of ((TOP-REAL 2) | Kb) by RELSET_1:22
.= Sq_Circ .: Kb by A5, RELAT_1:129
.= Cb by A3, Th23 ;
then  ex f1 being Function of ((TOP-REAL 2) | Kb),((TOP-REAL 2) | Cbb) st
( f00 = f1 & f1 is being_homeomorphism ) by A8, Th21, JGRAPH_1:46, TOPMETR:7;
hence  Cb is being_simple_closed_curve by Th24; ::_thesis: verum
end;

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
proof
let K0, C0 be Subset of (TOP-REAL 2); ::_thesis: ( 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  }  implies Sq_Circ " C0 c= K0 )
assume A1: ( 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  }  ) ; ::_thesis: Sq_Circ " C0 c= K0
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Sq_Circ " C0 or x in K0 )
assume A2: x in Sq_Circ " C0 ; ::_thesis: x in K0
then reconsider px = x as Point of (TOP-REAL 2) ;
set q = px;
A3: Sq_Circ . x in C0 by A2, FUNCT_1:def_7;
now__::_thesis:_(_(_px_=_0._(TOP-REAL_2)_&_-_1_<=_px_`1_&_px_`1_<=_1_&_-_1_<=_px_`2_&_px_`2_<=_1_)_or_(_px_<>_0._(TOP-REAL_2)_&_(_(_px_`2_<=_px_`1_&_-_(px_`1)_<=_px_`2_)_or_(_px_`2_>=_px_`1_&_px_`2_<=_-_(px_`1)_)_)_&_-_1_<=_px_`1_&_px_`1_<=_1_&_-_1_<=_px_`2_&_px_`2_<=_1_)_or_(_px_<>_0._(TOP-REAL_2)_&_not_(_px_`2_<=_px_`1_&_-_(px_`1)_<=_px_`2_)_&_not_(_px_`2_>=_px_`1_&_px_`2_<=_-_(px_`1)_)_&_-_1_<=_px_`1_&_px_`1_<=_1_&_-_1_<=_px_`2_&_px_`2_<=_1_)_)
percases ( px = 0. (TOP-REAL 2) or ( px <> 0. (TOP-REAL 2) & ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) ) or ( px <> 0. (TOP-REAL 2) & not ( px `2 <= px `1 & - (px `1) <= px `2 ) & not ( px `2 >= px `1 & px `2 <= - (px `1) ) ) ) ;
case px = 0. (TOP-REAL 2) ; ::_thesis: ( - 1 <= px `1 & px `1 <= 1 & - 1 <= px `2 & px `2 <= 1 )
hence  ( - 1 <= px `1 & px `1 <= 1 & - 1 <= px `2 & px `2 <= 1 ) by JGRAPH_2:3; ::_thesis: verum
end;
caseA4: ( px <> 0. (TOP-REAL 2) & ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) ) ; ::_thesis: ( - 1 <= px `1 & px `1 <= 1 & - 1 <= px `2 & px `2 <= 1 )
A5: now__::_thesis:_not_((px_`1)_^2)_+_((px_`2)_^2)_=_0
assume  ((px `1) ^2) + ((px `2) ^2) = 0 ; ::_thesis: contradiction
then  ( px `1 = 0 & px `2 = 0 ) by COMPLEX1:1;
hence  contradiction by A4, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A6: (px `1) ^2 >= 0 by XREAL_1:63;
A7: now__::_thesis:_not_px_`1_=_0
assume A8: px `1 = 0 ; ::_thesis: contradiction
then  px `2 = 0 by A4;
hence  contradiction by A4, A8, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A9: ( |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `1 = (px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2))) & |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `2 = (px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))) ) by EUCLID:52;
consider p1 being Point of (TOP-REAL 2) such that
A10: p1 = Sq_Circ . px and
A11: |.p1.| <= 1 by A1, A3;
 |.p1.| ^2 <= |.p1.| by A11, SQUARE_1:42;
then A12: |.p1.| ^2 <= 1 by A11, XXREAL_0:2;
A13: 1 + (((px `2) / (px `1)) ^2) > 0 by Lm1;
 Sq_Circ . px = |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| by A4, Def1;
then |.p1.| ^2 = (((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by A9, A10, JGRAPH_1:29
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by XCMPLX_1:76
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by XCMPLX_1:76
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by A13, SQUARE_1:def_2
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / (1 + (((px `2) / (px `1)) ^2))) by A13, SQUARE_1:def_2
.= (((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2)) by XCMPLX_1:62 ;
then  ((((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2))) * (1 + (((px `2) / (px `1)) ^2)) <= 1 * (1 + (((px `2) / (px `1)) ^2)) by A13, A12, XREAL_1:64;
then  ((px `1) ^2) + ((px `2) ^2) <= 1 + (((px `2) / (px `1)) ^2) by A13, XCMPLX_1:87;
then  ((px `1) ^2) + ((px `2) ^2) <= 1 + (((px `2) ^2) / ((px `1) ^2)) by XCMPLX_1:76;
then  (((px `1) ^2) + ((px `2) ^2)) - 1 <= (1 + (((px `2) ^2) / ((px `1) ^2))) - 1 by XREAL_1:9;
then  ((((px `1) ^2) + ((px `2) ^2)) - 1) * ((px `1) ^2) <= (((px `2) ^2) / ((px `1) ^2)) * ((px `1) ^2) by A6, XREAL_1:64;
then  (((px `1) ^2) * ((px `1) ^2)) + ((((px `2) ^2) - 1) * ((px `1) ^2)) <= (px `2) ^2 by A7, XCMPLX_1:6, XCMPLX_1:87;
then  (((((px `1) ^2) * ((px `1) ^2)) - (((px `1) ^2) * 1)) + (((px `1) ^2) * ((px `2) ^2))) - (1 * ((px `2) ^2)) <= 0 by XREAL_1:47;
then A14: (((px `1) ^2) - 1) * (((px `1) ^2) + ((px `2) ^2)) <= 0 ;
 (px `2) ^2 >= 0 by XREAL_1:63;
then A15: ((px `1) ^2) - 1 <= 0 by A6, A14, A5, XREAL_1:129;
then A16: px `1 <= 1 by SQUARE_1:43;
A17: - 1 <= px `1 by A15, SQUARE_1:43;
then  ( ( px `2 <= 1 & - (- (px `1)) >= - (px `2) ) or ( px `2 >= - 1 & - (px `2) >= - (- (px `1)) ) ) by A4, A16, XREAL_1:24, XXREAL_0:2;
then  ( ( px `2 <= 1 & 1 >= - (px `2) ) or ( px `2 >= - 1 & - (px `2) >= px `1 ) ) by A16, XXREAL_0:2;
then  ( ( px `2 <= 1 & - 1 <= - (- (px `2)) ) or ( px `2 >= - 1 & - (px `2) >= - 1 ) ) by A17, XREAL_1:24, XXREAL_0:2;
hence  ( - 1 <= px `1 & px `1 <= 1 & - 1 <= px `2 & px `2 <= 1 ) by A15, SQUARE_1:43, XREAL_1:24; ::_thesis: verum
end;
caseA18: ( px <> 0. (TOP-REAL 2) & not ( px `2 <= px `1 & - (px `1) <= px `2 ) & not ( px `2 >= px `1 & px `2 <= - (px `1) ) ) ; ::_thesis: ( - 1 <= px `1 & px `1 <= 1 & - 1 <= px `2 & px `2 <= 1 )
A19: now__::_thesis:_not_((px_`2)_^2)_+_((px_`1)_^2)_=_0
assume  ((px `2) ^2) + ((px `1) ^2) = 0 ; ::_thesis: contradiction
then  px `2 = 0 by COMPLEX1:1;
hence  contradiction by A18; ::_thesis: verum
end;
A20: (px `2) ^2 >= 0 by XREAL_1:63;
A21: px `2 <> 0 by A18;
A22: ( |[((px `1) / (sqrt (1 + (((px `1) / (px `2)) ^2)))),((px `2) / (sqrt (1 + (((px `1) / (px `2)) ^2))))]| `2 = (px `2) / (sqrt (1 + (((px `1) / (px `2)) ^2))) & |[((px `1) / (sqrt (1 + (((px `1) / (px `2)) ^2)))),((px `2) / (sqrt (1 + (((px `1) / (px `2)) ^2))))]| `1 = (px `1) / (sqrt (1 + (((px `1) / (px `2)) ^2))) ) by EUCLID:52;
consider p1 being Point of (TOP-REAL 2) such that
A23: p1 = Sq_Circ . px and
A24: |.p1.| <= 1 by A1, A3;
 |.p1.| ^2 <= |.p1.| by A24, SQUARE_1:42;
then A25: |.p1.| ^2 <= 1 by A24, XXREAL_0:2;
A26: 1 + (((px `1) / (px `2)) ^2) > 0 by Lm1;
 Sq_Circ . px = |[((px `1) / (sqrt (1 + (((px `1) / (px `2)) ^2)))),((px `2) / (sqrt (1 + (((px `1) / (px `2)) ^2))))]| by A18, Def1;
then |.p1.| ^2 = (((px `1) / (sqrt (1 + (((px `1) / (px `2)) ^2)))) ^2) + (((px `2) / (sqrt (1 + (((px `1) / (px `2)) ^2)))) ^2) by A22, A23, JGRAPH_1:29
.= (((px `2) ^2) / ((sqrt (1 + (((px `1) / (px `2)) ^2))) ^2)) + (((px `1) / (sqrt (1 + (((px `1) / (px `2)) ^2)))) ^2) by XCMPLX_1:76
.= (((px `2) ^2) / ((sqrt (1 + (((px `1) / (px `2)) ^2))) ^2)) + (((px `1) ^2) / ((sqrt (1 + (((px `1) / (px `2)) ^2))) ^2)) by XCMPLX_1:76
.= (((px `2) ^2) / (1 + (((px `1) / (px `2)) ^2))) + (((px `1) ^2) / ((sqrt (1 + (((px `1) / (px `2)) ^2))) ^2)) by A26, SQUARE_1:def_2
.= (((px `2) ^2) / (1 + (((px `1) / (px `2)) ^2))) + (((px `1) ^2) / (1 + (((px `1) / (px `2)) ^2))) by A26, SQUARE_1:def_2
.= (((px `2) ^2) + ((px `1) ^2)) / (1 + (((px `1) / (px `2)) ^2)) by XCMPLX_1:62 ;
then  ((((px `2) ^2) + ((px `1) ^2)) / (1 + (((px `1) / (px `2)) ^2))) * (1 + (((px `1) / (px `2)) ^2)) <= 1 * (1 + (((px `1) / (px `2)) ^2)) by A26, A25, XREAL_1:64;
then  ((px `2) ^2) + ((px `1) ^2) <= 1 + (((px `1) / (px `2)) ^2) by A26, XCMPLX_1:87;
then  ((px `2) ^2) + ((px `1) ^2) <= 1 + (((px `1) ^2) / ((px `2) ^2)) by XCMPLX_1:76;
then  (((px `2) ^2) + ((px `1) ^2)) - 1 <= (1 + (((px `1) ^2) / ((px `2) ^2))) - 1 by XREAL_1:9;
then  ((((px `2) ^2) + ((px `1) ^2)) - 1) * ((px `2) ^2) <= (((px `1) ^2) / ((px `2) ^2)) * ((px `2) ^2) by A20, XREAL_1:64;
then  (((px `2) ^2) * ((px `2) ^2)) + ((((px `1) ^2) - 1) * ((px `2) ^2)) <= (px `1) ^2 by A21, XCMPLX_1:6, XCMPLX_1:87;
then  (((((px `2) ^2) * ((px `2) ^2)) - (((px `2) ^2) * 1)) + (((px `2) ^2) * ((px `1) ^2))) - (1 * ((px `1) ^2)) <= 0 by XREAL_1:47;
then A27: (((px `2) ^2) - 1) * (((px `2) ^2) + ((px `1) ^2)) <= 0 ;
 (px `1) ^2 >= 0 by XREAL_1:63;
then A28: ((px `2) ^2) - 1 <= 0 by A20, A27, A19, XREAL_1:129;
then  ( - 1 <= px `2 & px `2 <= 1 ) by SQUARE_1:43;
then  ( ( px `1 <= 1 & 1 >= - (px `1) ) or ( px `1 >= - 1 & - (px `1) >= - 1 ) ) by A18, XXREAL_0:2;
then  ( ( px `1 <= 1 & - 1 <= - (- (px `1)) ) or ( px `1 >= - 1 & px `1 <= 1 ) ) by XREAL_1:24;
hence  ( - 1 <= px `1 & px `1 <= 1 & - 1 <= px `2 & px `2 <= 1 ) by A28, SQUARE_1:43; ::_thesis: verum
end;
end;
end;
hence  x in K0 by A1; ::_thesis: verum
end;

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))))]| ) )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( ( 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))))]| ) )
set q = p;
set px = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|;
A1: |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52;
A2: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
hereby  ::_thesis: ( ( ( ( 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))))]| ) )
assume A3: p = 0. (TOP-REAL 2) ; ::_thesis: (Sq_Circ ") . p = 0. (TOP-REAL 2)
then  Sq_Circ . p = p by Def1;
hence  (Sq_Circ ") . p = 0. (TOP-REAL 2) by A2, A3, FUNCT_1:34; ::_thesis: verum
end;
hereby  ::_thesis: ( ( 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))))]| )
A4: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
set q = p;
assume that
A5: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) and
A6: p <> 0. (TOP-REAL 2) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]|
set px = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]|;
A7: |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52;
A8: sqrt (1 + (((p `2) / (p `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A9: |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 = (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52;
then A10: (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2) / (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1) = (p `2) / (p `1) by A7, A8, XCMPLX_1:91;
then A11: (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2) / (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1)) ^2))) = p `2 by A9, A8, XCMPLX_1:89;
A12: now__::_thesis:_(_|[((p_`1)_*_(sqrt_(1_+_(((p_`2)_/_(p_`1))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`2)_/_(p_`1))_^2))))]|_`1_=_0_implies_not_|[((p_`1)_*_(sqrt_(1_+_(((p_`2)_/_(p_`1))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`2)_/_(p_`1))_^2))))]|_`2_=_0_)
assume  ( |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 = 0 & |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 = 0 ) ; ::_thesis: contradiction
then  ( p `1 = 0 & p `2 = 0 ) by A7, A9, A8, XCMPLX_1:6;
hence  contradiction by A6, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
 ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (- (p `1)) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ) ) by A5, A8, XREAL_1:64;
then  ( ( p `2 <= p `1 & (- (p `1)) * (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ) or ( |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 >= |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 & |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 <= - (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1) ) ) by A7, A9, A8, XREAL_1:64;
then  ( ( (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) & - (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1) <= |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 ) or ( |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 >= |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 & |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 <= - (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1) ) ) by A7, A8, EUCLID:52, XREAL_1:64;
then A13: Sq_Circ . |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| = |[((|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2) / (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1)) ^2)))),((|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2) / (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1)) ^2))))]| by A7, A9, A12, Def1, JGRAPH_2:3;
 (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2) / (|[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1)) ^2))) = p `1 by A7, A8, A10, XCMPLX_1:89;
then  p = Sq_Circ . |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A13, A11, EUCLID:53;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A4, FUNCT_1:34; ::_thesis: verum
end;
A14: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
A15: sqrt (1 + (((p `1) / (p `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A16: |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 = (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52;
then A17: (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1) / (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2) = (p `1) / (p `2) by A1, A15, XCMPLX_1:91;
then A18: (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1) / (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2)) ^2))) = p `1 by A16, A15, XCMPLX_1:89;
assume A19: ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
A20: now__::_thesis:_(_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_`2_=_0_implies_not_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_`1_=_0_)
assume that
A21: |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 = 0 and
 |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 = 0 ; ::_thesis: contradiction
 p `2 = 0 by A1, A15, A21, XCMPLX_1:6;
hence  contradiction by A19; ::_thesis: verum
end;
 ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by A19, JGRAPH_2:13;
then  ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (- (p `2)) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ) ) by A15, XREAL_1:64;
then  ( ( p `1 <= p `2 & (- (p `2)) * (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ) or ( |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 >= |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 & |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 <= - (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2) ) ) by A1, A16, A15, XREAL_1:64;
then  ( ( (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) & - (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2) <= |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 ) or ( |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 >= |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 & |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 <= - (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2) ) ) by A1, A15, EUCLID:52, XREAL_1:64;
then A22: Sq_Circ . |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| = |[((|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1) / (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2)) ^2)))),((|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1) / (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2)) ^2))))]| by A1, A16, A20, Th4, JGRAPH_2:3;
 (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1) / (|[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2)) ^2))) = p `2 by A1, A15, A17, XCMPLX_1:89;
then  p = Sq_Circ . |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A22, A18, EUCLID:53;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A14, FUNCT_1:34; ::_thesis: verum
end;

theorem Th29: :: JGRAPH_3:29
Sq_Circ " is Function of (TOP-REAL 2),(TOP-REAL 2)
proof
A1: the carrier of (TOP-REAL 2) c= rng Sq_Circ
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (TOP-REAL 2) or y in rng Sq_Circ )
assume  y in the carrier of (TOP-REAL 2) ; ::_thesis: y in rng Sq_Circ
then reconsider py = y as Point of (TOP-REAL 2) ;
A2: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
now__::_thesis:_(_(_py_=_0._(TOP-REAL_2)_&_ex_x_being_set_st_
(_x_in_dom_Sq_Circ_&_y_=_Sq_Circ_._x_)_)_or_(_(_(_py_`2_<=_py_`1_&_-_(py_`1)_<=_py_`2_)_or_(_py_`2_>=_py_`1_&_py_`2_<=_-_(py_`1)_)_)_&_py_<>_0._(TOP-REAL_2)_&_ex_x_being_set_st_
(_x_in_dom_Sq_Circ_&_y_=_Sq_Circ_._x_)_)_or_(_not_(_py_`2_<=_py_`1_&_-_(py_`1)_<=_py_`2_)_&_not_(_py_`2_>=_py_`1_&_py_`2_<=_-_(py_`1)_)_&_py_<>_0._(TOP-REAL_2)_&_ex_x_being_set_st_
(_x_in_dom_Sq_Circ_&_y_=_Sq_Circ_._x_)_)_)
percases ( py = 0. (TOP-REAL 2) or ( ( ( py `2 <= py `1 & - (py `1) <= py `2 ) or ( py `2 >= py `1 & py `2 <= - (py `1) ) ) & py <> 0. (TOP-REAL 2) ) or ( not ( py `2 <= py `1 & - (py `1) <= py `2 ) & not ( py `2 >= py `1 & py `2 <= - (py `1) ) & py <> 0. (TOP-REAL 2) ) ) ;
case py = 0. (TOP-REAL 2) ; ::_thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )
then  Sq_Circ . py = py by Def1;
hence  ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A2; ::_thesis: verum
end;
caseA3: ( ( ( py `2 <= py `1 & - (py `1) <= py `2 ) or ( py `2 >= py `1 & py `2 <= - (py `1) ) ) & py <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )
set q = py;
set px = |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]|;
A4: sqrt (1 + (((py `2) / (py `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A5: now__::_thesis:_(_|[((py_`1)_*_(sqrt_(1_+_(((py_`2)_/_(py_`1))_^2)))),((py_`2)_*_(sqrt_(1_+_(((py_`2)_/_(py_`1))_^2))))]|_`1_=_0_implies_not_|[((py_`1)_*_(sqrt_(1_+_(((py_`2)_/_(py_`1))_^2)))),((py_`2)_*_(sqrt_(1_+_(((py_`2)_/_(py_`1))_^2))))]|_`2_=_0_)
assume that
A6: |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1 = 0 and
A7: |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 = 0 ; ::_thesis: contradiction
 (py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))) = 0 by A7, EUCLID:52;
then A8: py `2 = 0 by A4, XCMPLX_1:6;
 (py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2))) = 0 by A6, EUCLID:52;
then  py `1 = 0 by A4, XCMPLX_1:6;
hence  contradiction by A3, A8, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A9: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
A10: |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1 = (py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2))) by EUCLID:52;
A11: |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 = (py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))) by EUCLID:52;
then A12: (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2) / (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1) = (py `2) / (py `1) by A10, A4, XCMPLX_1:91;
then A13: (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2) / (sqrt (1 + (((|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2) / (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1)) ^2))) = py `2 by A11, A4, XCMPLX_1:89;
 ( ( py `2 <= py `1 & - (py `1) <= py `2 ) or ( py `2 >= py `1 & (py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))) <= (- (py `1)) * (sqrt (1 + (((py `2) / (py `1)) ^2))) ) ) by A3, A4, XREAL_1:64;
then  ( ( py `2 <= py `1 & (- (py `1)) * (sqrt (1 + (((py `2) / (py `1)) ^2))) <= (py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))) ) or ( |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 >= |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1 & |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 <= - (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1) ) ) by A10, A11, A4, XREAL_1:64;
then  ( ( (py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))) <= (py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2))) & - (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1) <= |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 ) or ( |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 >= |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1 & |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 <= - (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1) ) ) by A10, A4, EUCLID:52, XREAL_1:64;
then A14: Sq_Circ . |[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| = |[((|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1) / (sqrt (1 + (((|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2) / (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1)) ^2)))),((|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2) / (sqrt (1 + (((|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2) / (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1)) ^2))))]| by A10, A11, A5, Def1, JGRAPH_2:3;
 (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1) / (sqrt (1 + (((|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2) / (|[((py `1) * (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) * (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1)) ^2))) = py `1 by A10, A4, A12, XCMPLX_1:89;
hence  ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A14, A13, A9, EUCLID:53; ::_thesis: verum
end;
caseA15: ( not ( py `2 <= py `1 & - (py `1) <= py `2 ) & not ( py `2 >= py `1 & py `2 <= - (py `1) ) & py <> 0. (TOP-REAL 2) ) ; ::_thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )
set q = py;
set px = |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]|;
A16: sqrt (1 + (((py `1) / (py `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A17: |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2 = (py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))) by EUCLID:52;
A18: now__::_thesis:_(_|[((py_`1)_*_(sqrt_(1_+_(((py_`1)_/_(py_`2))_^2)))),((py_`2)_*_(sqrt_(1_+_(((py_`1)_/_(py_`2))_^2))))]|_`2_=_0_implies_not_|[((py_`1)_*_(sqrt_(1_+_(((py_`1)_/_(py_`2))_^2)))),((py_`2)_*_(sqrt_(1_+_(((py_`1)_/_(py_`2))_^2))))]|_`1_=_0_)
assume that
A19: |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2 = 0 and
 |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1 = 0 ; ::_thesis: contradiction
 py `2 = 0 by A17, A16, A19, XCMPLX_1:6;
hence  contradiction by A15; ::_thesis: verum
end;
A20: |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1 = (py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2))) by EUCLID:52;
then A21: (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1) / (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2) = (py `1) / (py `2) by A17, A16, XCMPLX_1:91;
then A22: (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1) / (sqrt (1 + (((|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1) / (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2)) ^2))) = py `1 by A20, A16, XCMPLX_1:89;
 ( ( py `1 <= py `2 & - (py `2) <= py `1 ) or ( py `1 >= py `2 & py `1 <= - (py `2) ) ) by A15, JGRAPH_2:13;
then  ( ( py `1 <= py `2 & - (py `2) <= py `1 ) or ( py `1 >= py `2 & (py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2))) <= (- (py `2)) * (sqrt (1 + (((py `1) / (py `2)) ^2))) ) ) by A16, XREAL_1:64;
then  ( ( py `1 <= py `2 & (- (py `2)) * (sqrt (1 + (((py `1) / (py `2)) ^2))) <= (py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2))) ) or ( |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1 >= |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2 & |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1 <= - (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2) ) ) by A17, A20, A16, XREAL_1:64;
then  ( ( (py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2))) <= (py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))) & - (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2) <= |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1 ) or ( |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1 >= |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2 & |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1 <= - (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2) ) ) by A17, A16, EUCLID:52, XREAL_1:64;
then A23: Sq_Circ . |[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| = |[((|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1) / (sqrt (1 + (((|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1) / (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2)) ^2)))),((|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2) / (sqrt (1 + (((|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1) / (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2)) ^2))))]| by A17, A20, A18, Th4, JGRAPH_2:3;
A24: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
 (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2) / (sqrt (1 + (((|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `1) / (|[((py `1) * (sqrt (1 + (((py `1) / (py `2)) ^2)))),((py `2) * (sqrt (1 + (((py `1) / (py `2)) ^2))))]| `2)) ^2))) = py `2 by A17, A16, A21, XCMPLX_1:89;
hence  ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A23, A22, A24, EUCLID:53; ::_thesis: verum
end;
end;
end;
hence  y in rng Sq_Circ by FUNCT_1:def_3; ::_thesis: verum
end;
A25: rng (Sq_Circ ") = dom Sq_Circ by FUNCT_1:33
.= the carrier of (TOP-REAL 2) by FUNCT_2:def_1 ;
 dom (Sq_Circ ") = rng Sq_Circ by FUNCT_1:33;
then  dom (Sq_Circ ") = the carrier of (TOP-REAL 2) by A1, XBOOLE_0:def_10;
hence  Sq_Circ " is Function of (TOP-REAL 2),(TOP-REAL 2) by A25, FUNCT_2:1; ::_thesis: verum
end;

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))))]| ) )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p <> 0. (TOP-REAL 2) implies ( ( ( ( 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))))]| ) ) )
A1: ( - (p `2) < p `1 implies - (- (p `2)) > - (p `1) ) by XREAL_1:24;
assume A2: p <> 0. (TOP-REAL 2) ; ::_thesis: ( ( ( ( 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))))]| ) )
hereby  ::_thesis: ( ( 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))))]| )
assume A3: ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
now__::_thesis:_(_(_p_`1_<=_p_`2_&_-_(p_`2)_<=_p_`1_&_(Sq_Circ_")_._p_=_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_or_(_p_`1_>=_p_`2_&_p_`1_<=_-_(p_`2)_&_(Sq_Circ_")_._p_=_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_)
percases ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by A3;
caseA4: ( p `1 <= p `2 & - (p `2) <= p `1 ) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
now__::_thesis:_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_implies_(Sq_Circ_")_._p_=_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)
assume A5: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
A6: now__::_thesis:_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_)
percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A5;
case ( p `2 <= p `1 & - (p `1) <= p `2 ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
hence  ( p `1 = p `2 or p `1 = - (p `2) ) by A4, XXREAL_0:1; ::_thesis: verum
end;
case ( p `2 >= p `1 & p `2 <= - (p `1) ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
then  - (p `2) >= - (- (p `1)) by XREAL_1:24;
hence  ( p `1 = p `2 or p `1 = - (p `2) ) by A4, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_p_`1_=_p_`2_&_(Sq_Circ_")_._p_=_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_or_(_p_`1_=_-_(p_`2)_&_(Sq_Circ_")_._p_=_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_)
percases ( p `1 = p `2 or p `1 = - (p `2) ) by A6;
case p `1 = p `2 ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, A5, Th28; ::_thesis: verum
end;
case p `1 = - (p `2) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
then  ( p `1 <> 0 & - (p `1) = p `2 ) by A2, EUCLID:53, EUCLID:54;
then  ( (p `1) / (p `2) = - 1 & (p `2) / (p `1) = - 1 ) by XCMPLX_1:197, XCMPLX_1:198;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, A5, Th28; ::_thesis: verum
end;
end;
end;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ; ::_thesis: verum
end;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by Th28; ::_thesis: verum
end;
caseA7: ( p `1 >= p `2 & p `1 <= - (p `2) ) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
now__::_thesis:_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_implies_(Sq_Circ_")_._p_=_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)
assume A8: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
A9: now__::_thesis:_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_)
percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A8;
case ( p `2 <= p `1 & - (p `1) <= p `2 ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
then  - (- (p `1)) >= - (p `2) by XREAL_1:24;
hence  ( p `1 = p `2 or p `1 = - (p `2) ) by A7, XXREAL_0:1; ::_thesis: verum
end;
case ( p `2 >= p `1 & p `2 <= - (p `1) ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
hence  ( p `1 = p `2 or p `1 = - (p `2) ) by A7, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_p_`1_=_p_`2_&_(Sq_Circ_")_._p_=_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_or_(_p_`1_=_-_(p_`2)_&_(Sq_Circ_")_._p_=_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_)_)
percases ( p `1 = p `2 or p `1 = - (p `2) ) by A9;
case p `1 = p `2 ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, A8, Th28; ::_thesis: verum
end;
caseA10: p `1 = - (p `2) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|
then  ( p `1 <> 0 & - (p `1) = p `2 ) by A2, EUCLID:53, EUCLID:54;
then A11: (p `2) / (p `1) = - 1 by XCMPLX_1:197;
 p `2 <> 0 by A2, A10, EUCLID:53, EUCLID:54;
then  (p `1) / (p `2) = - 1 by A10, XCMPLX_1:197;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A2, A8, A11, Th28; ::_thesis: verum
end;
end;
end;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ; ::_thesis: verum
end;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by Th28; ::_thesis: verum
end;
end;
end;
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| ; ::_thesis: verum
end;
A12: ( - (p `2) > p `1 implies - (- (p `2)) < - (p `1) ) by XREAL_1:24;
assume  ( not ( p `1 <= p `2 & - (p `2) <= p `1 ) & not ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ; ::_thesis: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]|
hence  (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A2, A1, A12, Th28; ::_thesis: verum
end;

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 )
proof
let X be non empty TopSpace; ::_thesis: 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 )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( 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 * (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: ( f2 is continuous & ( 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 * (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous )
consider g2 being Function of X,R^1 such that
A3: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = sqrt (1 + ((r1 / r2) ^2)) and
A4: g2 is continuous by A1, A2, Th8;
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 & g2 . p = r0 holds
g3 . p = r1 * r0 and
A6: g3 is continuous by A1, A4, JGRAPH_2:25;
 for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = r1 * (sqrt (1 + ((r1 / r2) ^2)))
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = r1 * (sqrt (1 + ((r1 / r2) ^2)))
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = r1 * (sqrt (1 + ((r1 / r2) ^2))) )
assume that
A7: f1 . p = r1 and
A8: f2 . p = r2 ; ::_thesis: g3 . p = r1 * (sqrt (1 + ((r1 / r2) ^2)))
 g2 . p = sqrt (1 + ((r1 / r2) ^2)) by A3, A7, A8;
hence  g3 . p = r1 * (sqrt (1 + ((r1 / r2) ^2))) by A5, A7; ::_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 * (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) by A6; ::_thesis: verum
end;

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 )
proof
let X be non empty TopSpace; ::_thesis: 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 )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( 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 (1 + ((r1 / r2) ^2))) ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: f2 is continuous 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 = r2 * (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous )
consider g2 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
g2 . p = sqrt (1 + ((r1 / r2) ^2)) and
A5: g2 is continuous by A1, A2, A3, Th8;
consider g3 being Function of X,R^1 such that
A6: for p being Point of X
for r2, r0 being real number st f2 . p = r2 & g2 . p = r0 holds
g3 . p = r2 * r0 and
A7: g3 is continuous by A2, A5, JGRAPH_2:25;
 for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = r2 * (sqrt (1 + ((r1 / r2) ^2)))
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = r2 * (sqrt (1 + ((r1 / r2) ^2)))
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = r2 * (sqrt (1 + ((r1 / r2) ^2))) )
assume that
A8: f1 . p = r1 and
A9: f2 . p = r2 ; ::_thesis: g3 . p = r2 * (sqrt (1 + ((r1 / r2) ^2)))
 g2 . p = sqrt (1 + ((r1 / r2) ^2)) by A4, A8, A9;
hence  g3 . p = r2 * (sqrt (1 + ((r1 / r2) ^2))) by A6, A9; ::_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 (1 + ((r1 / r2) ^2))) ) & g is continuous ) by A7; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( 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 ) implies f is continuous )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
A1: 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))) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ; ::_thesis: f is continuous
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
 for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0
 q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj1 . q by Lm6
.= q2 `1 by PSCOMP_1:def_5 ;
hence  g1 . q <> 0 by A2; ::_thesis: verum
end;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: 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 (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th32;
A6: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
f_._x_=_g3_._x
let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x )
assume A7: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
 x in the carrier of ((TOP-REAL 2) | K1) by A7;
then  x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
 f . r = (r `1) * (sqrt (1 + (((r `2) / (r `1)) ^2))) by A1, A7;
hence  f . x = g3 . x by A4, A9, A8; ::_thesis: verum
end;
 dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then  dom f = dom g3 by FUNCT_2:def_1;
hence  f is continuous by A5, A6, FUNCT_1:2; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( 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 ) implies f is continuous )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
A1: 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))) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ; ::_thesis: f is continuous
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
 for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0
 q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj1 . q by Lm6
.= q2 `1 by PSCOMP_1:def_5 ;
hence  g1 . q <> 0 by A2; ::_thesis: verum
end;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: 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 = r1 * (sqrt (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th31;
A6: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
f_._x_=_g3_._x
let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x )
assume A7: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
 x in the carrier of ((TOP-REAL 2) | K1) by A7;
then  x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
 f . r = (r `2) * (sqrt (1 + (((r `2) / (r `1)) ^2))) by A1, A7;
hence  f . x = g3 . x by A4, A9, A8; ::_thesis: verum
end;
 dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then  dom f = dom g3 by FUNCT_2:def_1;
hence  f is continuous by A5, A6, FUNCT_1:2; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( 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 ) implies f is continuous )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
A1: 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))) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ; ::_thesis: f is continuous
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
 for q being Point of ((TOP-REAL 2) | K1) holds g2 . q <> 0
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g2 . q <> 0
 q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g2 . q = proj2 . q by Lm4
.= q2 `2 by PSCOMP_1:def_6 ;
hence  g2 . q <> 0 by A2; ::_thesis: verum
end;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being real number st g1 . q = r1 & g2 . q = r2 holds
g3 . q = r2 * (sqrt (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th32;
A6: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
f_._x_=_g3_._x
let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x )
assume A7: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
 x in the carrier of ((TOP-REAL 2) | K1) by A7;
then  x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
 f . r = (r `2) * (sqrt (1 + (((r `1) / (r `2)) ^2))) by A1, A7;
hence  f . x = g3 . x by A4, A9, A8; ::_thesis: verum
end;
 dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then  dom f = dom g3 by FUNCT_2:def_1;
hence  f is continuous by A5, A6, FUNCT_1:2; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( 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 ) implies f is continuous )
reconsider g1 = proj1 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm7;
reconsider g2 = proj2 | K1 as continuous Function of ((TOP-REAL 2) | K1),R^1 by Lm5;
assume that
A1: 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))) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ; ::_thesis: f is continuous
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
 for q being Point of ((TOP-REAL 2) | K1) holds g2 . q <> 0
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g2 . q <> 0
 q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g2 . q = proj2 . q by Lm4
.= q2 `2 by PSCOMP_1:def_6 ;
hence  g2 . q <> 0 by A2; ::_thesis: verum
end;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A4: for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being real number st g1 . q = r1 & g2 . q = r2 holds
g3 . q = r1 * (sqrt (1 + ((r1 / r2) ^2))) and
A5: g3 is continuous by Th31;
A6: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
f_._x_=_g3_._x
let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x )
assume A7: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
 x in the carrier of ((TOP-REAL 2) | K1) by A7;
then  x in K1 by PRE_TOPC:8;
then reconsider r = x as Point of (TOP-REAL 2) ;
A8: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A9: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by Lm4, Lm6;
 f . r = (r `1) * (sqrt (1 + (((r `1) / (r `2)) ^2))) by A1, A7;
hence  f . x = g3 . x by A4, A9, A8; ::_thesis: verum
end;
 dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then  dom f = dom g3 by FUNCT_2:def_1;
hence  f is continuous by A5, A6, FUNCT_1:2; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: proj2 * ((Sq_Circ ") | K1) is Function of ((TOP-REAL 2) | K1),R^1
A1: rng (proj2 * ((Sq_Circ ") | K1)) c= rng proj2 by RELAT_1:26;
A2: dom ((Sq_Circ ") | K1) c= dom (proj2 * ((Sq_Circ ") | K1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((Sq_Circ ") | K1) or x in dom (proj2 * ((Sq_Circ ") | K1)) )
A3: rng (Sq_Circ ") c= the carrier of (TOP-REAL 2) by Th29, RELAT_1:def_19;
assume A4: x in dom ((Sq_Circ ") | K1) ; ::_thesis: x in dom (proj2 * ((Sq_Circ ") | K1))
then  x in (dom (Sq_Circ ")) /\ K1 by RELAT_1:61;
then  x in dom (Sq_Circ ") by XBOOLE_0:def_4;
then A5: (Sq_Circ ") . x in rng (Sq_Circ ") by FUNCT_1:3;
 ((Sq_Circ ") | K1) . x = (Sq_Circ ") . x by A4, FUNCT_1:47;
hence  x in dom (proj2 * ((Sq_Circ ") | K1)) by A4, A5, A3, Lm3, FUNCT_1:11; ::_thesis: verum
end;
 dom (proj2 * ((Sq_Circ ") | K1)) c= dom ((Sq_Circ ") | K1) by RELAT_1:25;
then  dom (proj2 * ((Sq_Circ ") | K1)) = dom ((Sq_Circ ") | K1) by A2, XBOOLE_0:def_10
.= (dom (Sq_Circ ")) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by Th29, FUNCT_2:def_1
.= K1 by XBOOLE_1:28
.= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ;
hence  proj2 * ((Sq_Circ ") | K1) is Function of ((TOP-REAL 2) | K1),R^1 by A1, FUNCT_2:2, TOPMETR:17, XBOOLE_1:1; ::_thesis: verum
end;

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
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: proj1 * ((Sq_Circ ") | K1) is Function of ((TOP-REAL 2) | K1),R^1
A1: rng (proj1 * ((Sq_Circ ") | K1)) c= rng proj1 by RELAT_1:26;
A2: dom ((Sq_Circ ") | K1) c= dom (proj1 * ((Sq_Circ ") | K1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((Sq_Circ ") | K1) or x in dom (proj1 * ((Sq_Circ ") | K1)) )
A3: rng (Sq_Circ ") c= the carrier of (TOP-REAL 2) by Th29, RELAT_1:def_19;
assume A4: x in dom ((Sq_Circ ") | K1) ; ::_thesis: x in dom (proj1 * ((Sq_Circ ") | K1))
then  x in (dom (Sq_Circ ")) /\ K1 by RELAT_1:61;
then  x in dom (Sq_Circ ") by XBOOLE_0:def_4;
then A5: (Sq_Circ ") . x in rng (Sq_Circ ") by FUNCT_1:3;
 ((Sq_Circ ") | K1) . x = (Sq_Circ ") . x by A4, FUNCT_1:47;
hence  x in dom (proj1 * ((Sq_Circ ") | K1)) by A4, A5, A3, Lm2, FUNCT_1:11; ::_thesis: verum
end;
 dom (proj1 * ((Sq_Circ ") | K1)) c= dom ((Sq_Circ ") | K1) by RELAT_1:25;
then  dom (proj1 * ((Sq_Circ ") | K1)) = dom ((Sq_Circ ") | K1) by A2, XBOOLE_0:def_10
.= (dom (Sq_Circ ")) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by Th29, FUNCT_2:def_1
.= K1 by XBOOLE_1:28
.= the carrier of ((TOP-REAL 2) | K1) by PRE_TOPC:8 ;
hence  proj1 * ((Sq_Circ ") | K1) is Function of ((TOP-REAL 2) | K1),R^1 by A1, FUNCT_2:2, TOPMETR:17, XBOOLE_1:1; ::_thesis: verum
end;

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
proof
let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( 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) )  }  implies f is continuous )
assume A1: ( 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) )  }  ) ; ::_thesis: f is continuous
then  1.REAL 2 in K0 by Lm9, Lm10;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
reconsider g1 = proj1 * ((Sq_Circ ") | K1) as Function of ((TOP-REAL 2) | K1),R^1 by Lm18;
 for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
g1 . p = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))
proof
A2: dom ((Sq_Circ ") | K1) = (dom (Sq_Circ ")) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by Th29, FUNCT_2:def_1
.= K1 by XBOOLE_1:28 ;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) )
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
assume A4: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))
then  ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A3;
then A5: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by Th28;
 ((Sq_Circ ") | K1) . p = (Sq_Circ ") . p by A4, A3, FUNCT_1:49;
then g1 . p = proj1 . |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A4, A2, A3, A5, FUNCT_1:13
.= |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 by PSCOMP_1:def_5
.= (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52 ;
hence  g1 . p = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ; ::_thesis: verum
end;
then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that
A6: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f1 . p = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ;
reconsider g2 = proj2 * ((Sq_Circ ") | K1) as Function of ((TOP-REAL 2) | K1),R^1 by Lm17;
 for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
g2 . p = (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2)))
proof
A7: dom ((Sq_Circ ") | K1) = (dom (Sq_Circ ")) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by Th29, FUNCT_2:def_1
.= K1 by XBOOLE_1:28 ;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) )
A8: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
assume A9: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2)))
then  ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A8;
then A10: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by Th28;
 ((Sq_Circ ") | K1) . p = (Sq_Circ ") . p by A9, A8, FUNCT_1:49;
then g2 . p = proj2 . |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by A9, A7, A8, A10, FUNCT_1:13
.= |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 by PSCOMP_1:def_6
.= (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52 ;
hence  g2 . p = (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ; ::_thesis: verum
end;
then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that
A11: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f2 . p = (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ;
A12: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies q `1 <> 0 )
A13: 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
then A14: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A13;
now__::_thesis:_not_q_`1_=_0
assume A15: q `1 = 0 ; ::_thesis: contradiction
then  q `2 = 0 by A14;
hence  contradiction by A14, A15, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence  q `1 <> 0 ; ::_thesis: verum
end;
then A16: f1 is continuous by A6, Th33;
A17: now__::_thesis:_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]|
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
A18: |[x,y]| in K1 and
A19: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[r,s]|
set p99 = |[x,y]|;
A20: ex p3 being Point of (TOP-REAL 2) st
( |[x,y]| = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A18;
A21: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
then A22: f1 . |[x,y]| = (|[x,y]| `1) * (sqrt (1 + (((|[x,y]| `2) / (|[x,y]| `1)) ^2))) by A6, A18;
((Sq_Circ ") | K0) . |[x,y]| = (Sq_Circ ") . |[x,y]| by A18, FUNCT_1:49
.= |[((|[x,y]| `1) * (sqrt (1 + (((|[x,y]| `2) / (|[x,y]| `1)) ^2)))),((|[x,y]| `2) * (sqrt (1 + (((|[x,y]| `2) / (|[x,y]| `1)) ^2))))]| by A20, Th28
.= |[r,s]| by A11, A18, A19, A21, A22 ;
hence  f . |[x,y]| = |[r,s]| by A1; ::_thesis: verum
end;
 f2 is continuous by A12, A11, Th34;
hence  f is continuous by A1, A16, A17, Lm13, JGRAPH_2:35; ::_thesis: verum
end;

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
proof
let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: 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
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( 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) )  }  implies f is continuous )
assume A1: ( 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) )  }  ) ; ::_thesis: f is continuous
then  1.REAL 2 in K0 by Lm14, Lm15;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
reconsider g1 = proj2 * ((Sq_Circ ") | K1) as Function of ((TOP-REAL 2) | K1),R^1 by Lm17;
 for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
g1 . p = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2)))
proof
A2: dom ((Sq_Circ ") | K1) = (dom (Sq_Circ ")) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by Th29, FUNCT_2:def_1
.= K1 by XBOOLE_1:28 ;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) )
A3: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
assume A4: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2)))
then  ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A3;
then A5: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by Th30;
 ((Sq_Circ ") | K1) . p = (Sq_Circ ") . p by A4, A3, FUNCT_1:49;
then g1 . p = proj2 . |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A4, A2, A3, A5, FUNCT_1:13
.= |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 by PSCOMP_1:def_6
.= (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52 ;
hence  g1 . p = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ; ::_thesis: verum
end;
then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that
A6: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f1 . p = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ;
reconsider g2 = proj1 * ((Sq_Circ ") | K1) as Function of ((TOP-REAL 2) | K1),R^1 by Lm18;
 for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
g2 . p = (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))
proof
A7: dom ((Sq_Circ ") | K1) = (dom (Sq_Circ ")) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by Th29, FUNCT_2:def_1
.= K1 by XBOOLE_1:28 ;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) )
A8: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
assume A9: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))
then  ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A8;
then A10: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by Th30;
 ((Sq_Circ ") | K1) . p = (Sq_Circ ") . p by A9, A8, FUNCT_1:49;
then g2 . p = proj1 . |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by A9, A7, A8, A10, FUNCT_1:13
.= |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 by PSCOMP_1:def_5
.= (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52 ;
hence  g2 . p = (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ; ::_thesis: verum
end;
then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that
A11: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f2 . p = (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ;
A12: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies q `2 <> 0 )
A13: 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
then A14: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A13;
now__::_thesis:_not_q_`2_=_0
assume A15: q `2 = 0 ; ::_thesis: contradiction
then  q `1 = 0 by A14;
hence  contradiction by A14, A15, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence  q `2 <> 0 ; ::_thesis: verum
end;
then A16: f1 is continuous by A6, Th35;
A17: 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
A18: |[x,y]| in K1 and
A19: ( s = f2 . |[x,y]| & r = f1 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[s,r]|
set p99 = |[x,y]|;
A20: ex p3 being Point of (TOP-REAL 2) st
( |[x,y]| = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A1, A18;
A21: the carrier of ((TOP-REAL 2) | K1) = K1 by PRE_TOPC:8;
then A22: f1 . |[x,y]| = (|[x,y]| `2) * (sqrt (1 + (((|[x,y]| `1) / (|[x,y]| `2)) ^2))) by A6, A18;
((Sq_Circ ") | K0) . |[x,y]| = (Sq_Circ ") . |[x,y]| by A18, FUNCT_1:49
.= |[((|[x,y]| `1) * (sqrt (1 + (((|[x,y]| `1) / (|[x,y]| `2)) ^2)))),((|[x,y]| `2) * (sqrt (1 + (((|[x,y]| `1) / (|[x,y]| `2)) ^2))))]| by A20, Th30
.= |[s,r]| by A11, A18, A19, A21, A22 ;
hence  f . |[x,y]| = |[s,r]| by A1; ::_thesis: verum
end;
 f2 is continuous by A12, A11, Th36;
hence  f is continuous by A1, A16, A17, Lm13, JGRAPH_2:35; ::_thesis: verum
end;

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 )
proof
reconsider K5 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1)  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:47;
reconsider K4 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
reconsider K3 =  {  p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1) <= p7 `2  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:47;
reconsider K2 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ) );
set b0 = NonZero (TOP-REAL 2);
defpred S2[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ) );
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 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 )
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 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 )
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( 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) )  }  implies ( f is continuous & K0 is closed ) )
set k0 =  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  ;
assume that
A1: f = (Sq_Circ ") | K0 and
A2: ( B0 = NonZero (TOP-REAL 2) & K0 =  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  ) ; ::_thesis: ( f is continuous & K0 is closed )
 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;
  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  c= NonZero (TOP-REAL 2) from JGRAPH_3:sch_1();
then A3: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by A2, PRE_TOPC:7;
reconsider K1 =  {  p7 where p7 is Point of (TOP-REAL 2) : S1[p7]  }  as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
A4: (K2 /\ K3) \/ (K4 /\ K5) c= K1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A5: x in (K2 /\ K3) \/ (K4 /\ K5) ; ::_thesis: x in K1
percases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A5, XBOOLE_0:def_3;
supposeA6: x in K2 /\ K3 ; ::_thesis: x in K1
then  x in K3 by XBOOLE_0:def_4;
then A7: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & - (p8 `1) <= p8 `2 ) ;
 x in K2 by A6, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `2 <= p7 `1 ) ;
hence  x in K1 by A7; ::_thesis: verum
end;
supposeA8: x in K4 /\ K5 ; ::_thesis: x in K1
then  x in K5 by XBOOLE_0:def_4;
then A9: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & p8 `2 <= - (p8 `1) ) ;
 x in K4 by A8, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `2 >= p7 `1 ) ;
hence  x in K1 by A9; ::_thesis: verum
end;
end;
end;
A10: ( K2 /\ K3 is closed & K4 /\ K5 is closed ) by TOPS_1:8;
 K1 c= (K2 /\ K3) \/ (K4 /\ K5)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) )
assume  x in K1 ; ::_thesis: x in (K2 /\ K3) \/ (K4 /\ K5)
then  ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ) ;
then  ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ;
then  ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def_4;
hence  x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def_3; ::_thesis: verum
end;
then  K1 = (K2 /\ K3) \/ (K4 /\ K5) by A4, XBOOLE_0:def_10;
then A11: K1 is closed by A10, TOPS_1:9;
  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  =  {  p7 where p7 is Point of (TOP-REAL 2) : S2[p7]  }  /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch_2();
then  K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) by A2, PRE_TOPC:def_5;
hence  ( f is continuous & K0 is closed ) by A1, A2, A3, A11, Th37, PRE_TOPC:13; ::_thesis: verum
end;

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 )
proof
reconsider K5 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= - (p7 `2)  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:48;
reconsider K4 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
reconsider K3 =  {  p7 where p7 is Point of (TOP-REAL 2) : - (p7 `2) <= p7 `1  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:48;
reconsider K2 =  {  p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2  }  as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `1 <= $1 `2 & - ($1 `2) <= $1 `1 ) or ( $1 `1 >= $1 `2 & $1 `1 <= - ($1 `2) ) );
defpred S2[ Point of (TOP-REAL 2)] means ( ( $1 `1 <= $1 `2 & - ($1 `2) <= $1 `1 ) or ( $1 `1 >= $1 `2 & $1 `1 <= - ($1 `2) ) );
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 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 )
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 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 )
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( 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) )  }  implies ( f is continuous & K0 is closed ) )
set k0 =  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  ;
set b0 = NonZero (TOP-REAL 2);
assume that
A1: f = (Sq_Circ ") | K0 and
A2: ( B0 = NonZero (TOP-REAL 2) & K0 =  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  ) ; ::_thesis: ( f is continuous & K0 is closed )
 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;
  {  p where p is Point of (TOP-REAL 2) : ( S1[p] & p <> 0. (TOP-REAL 2) )  }  c= NonZero (TOP-REAL 2) from JGRAPH_3:sch_1();
then A3: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by A2, PRE_TOPC:7;
set k1 =  {  p7 where p7 is Point of (TOP-REAL 2) : S2[p7]  }  ;
A4: ( K2 /\ K3 is closed & K4 /\ K5 is closed ) by TOPS_1:8;
reconsider K1 =  {  p7 where p7 is Point of (TOP-REAL 2) : S2[p7]  }  as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
A5: (K2 /\ K3) \/ (K4 /\ K5) c= K1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A6: x in (K2 /\ K3) \/ (K4 /\ K5) ; ::_thesis: x in K1
percases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A6, XBOOLE_0:def_3;
supposeA7: x in K2 /\ K3 ; ::_thesis: x in K1
then  x in K3 by XBOOLE_0:def_4;
then A8: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & - (p8 `2) <= p8 `1 ) ;
 x in K2 by A7, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `1 <= p7 `2 ) ;
hence  x in K1 by A8; ::_thesis: verum
end;
supposeA9: x in K4 /\ K5 ; ::_thesis: x in K1
then  x in K5 by XBOOLE_0:def_4;
then A10: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & p8 `1 <= - (p8 `2) ) ;
 x in K4 by A9, XBOOLE_0:def_4;
then  ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `1 >= p7 `2 ) ;
hence  x in K1 by A10; ::_thesis: verum
end;
end;
end;
 K1 c= (K2 /\ K3) \/ (K4 /\ K5)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) )
assume  x in K1 ; ::_thesis: x in (K2 /\ K3) \/ (K4 /\ K5)
then  ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ) ;
then  ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ;
then  ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def_4;
hence  x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def_3; ::_thesis: verum
end;
then  K1 = (K2 /\ K3) \/ (K4 /\ K5) by A5, XBOOLE_0:def_10;
then A11: K1 is closed by A4, TOPS_1:9;
  {  p where p is Point of (TOP-REAL 2) : ( S2[p] & p <> 0. (TOP-REAL 2) )  }  =  {  p7 where p7 is Point of (TOP-REAL 2) : S2[p7]  }  /\ (NonZero (TOP-REAL 2)) from JGRAPH_3:sch_2();
then  K0 = K1 /\ ([#] ((TOP-REAL 2) | B0)) by A2, PRE_TOPC:def_5;
hence  ( f is continuous & K0 is closed ) by A1, A2, A3, A11, Th38, PRE_TOPC:13; ::_thesis: verum
end;

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 )
proof
set Y1 = |[(- 1),1]|;
set B0 = {(0. (TOP-REAL 2))};
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( D ` = {(0. (TOP-REAL 2))} implies ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = (Sq_Circ ") | D & h is continuous ) )
A1: the carrier of ((TOP-REAL 2) | D) = D by PRE_TOPC:8;
 dom (Sq_Circ ") = the carrier of (TOP-REAL 2) by Th29, FUNCT_2:def_1;
then A2: dom ((Sq_Circ ") | D) = the carrier of (TOP-REAL 2) /\ D by RELAT_1:61
.= the carrier of ((TOP-REAL 2) | D) by A1, XBOOLE_1:28 ;
assume A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = (Sq_Circ ") | D & h is continuous )
then A4: D = {(0. (TOP-REAL 2))} `
.= NonZero (TOP-REAL 2) by SUBSET_1:def_4 ;
A5:  {  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) )  }  c= the carrier of ((TOP-REAL 2) | D)
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 `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) )  }  or x in the carrier of ((TOP-REAL 2) | D) )
assume  x in  {  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) )  }  ; ::_thesis: x in the carrier of ((TOP-REAL 2) | D)
then A6: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) ;
now__::_thesis:_x_in_D
assume  not x in D ; ::_thesis: contradiction
then  x in the carrier of (TOP-REAL 2) \ D by A6, XBOOLE_0:def_5;
then  x in D ` by SUBSET_1:def_4;
hence  contradiction by A3, A6, TARSKI:def_1; ::_thesis: verum
end;
hence  x in the carrier of ((TOP-REAL 2) | D) by PRE_TOPC:8; ::_thesis: verum
end;
 1.REAL 2 in  {  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) )  }  by Lm9, Lm10;
then reconsider 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) )  }  as non empty Subset of ((TOP-REAL 2) | D) by A5;
A7: K0 = the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8;
A8:  {  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) )  }  c= the carrier of ((TOP-REAL 2) | D)
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 `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) )  }  or x in the carrier of ((TOP-REAL 2) | D) )
assume  x in  {  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) )  }  ; ::_thesis: x in the carrier of ((TOP-REAL 2) | D)
then A9: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) ;
now__::_thesis:_x_in_D
assume  not x in D ; ::_thesis: contradiction
then  x in the carrier of (TOP-REAL 2) \ D by A9, XBOOLE_0:def_5;
then  x in D ` by SUBSET_1:def_4;
hence  contradiction by A3, A9, TARSKI:def_1; ::_thesis: verum
end;
hence  x in the carrier of ((TOP-REAL 2) | D) by PRE_TOPC:8; ::_thesis: verum
end;
 ( |[(- 1),1]| `1 = - 1 & |[(- 1),1]| `2 = 1 ) by EUCLID:52;
then  |[(- 1),1]| in  {  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) )  }  by JGRAPH_2:3;
then reconsider K1 =  {  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) )  }  as non empty Subset of ((TOP-REAL 2) | D) by A8;
A10: K1 = the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8;
A11: 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 A12: 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 A4, A12, XBOOLE_0:def_5;
then  ( ( ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) or ( ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1, XREAL_1:26;
then  ( x in K0 or x in K1 ) ;
hence  x in K0 \/ K1 by XBOOLE_0:def_3; ::_thesis: verum
end;
A13: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= NonZero (TOP-REAL 2) by A4, PRE_TOPC:def_5 ;
A14: 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 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
hence  z in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
A15: rng ((Sq_Circ ") | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
proof
reconsider K00 = K0 as Subset of (TOP-REAL 2) by A14;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((Sq_Circ ") | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) )
A16: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K00) holds
q `1 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K00) implies q `1 <> 0 )
A17: the carrier of ((TOP-REAL 2) | K00) = K0 by PRE_TOPC:8;
assume  q in the carrier of ((TOP-REAL 2) | K00) ; ::_thesis: q `1 <> 0
then A18: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A17;
now__::_thesis:_not_q_`1_=_0
assume A19: q `1 = 0 ; ::_thesis: contradiction
then  q `2 = 0 by A18;
hence  contradiction by A18, A19, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence  q `1 <> 0 ; ::_thesis: verum
end;
assume  y in rng ((Sq_Circ ") | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0)
then consider x being set  such that
A20: x in dom ((Sq_Circ ") | K0) and
A21: y = ((Sq_Circ ") | K0) . x by FUNCT_1:def_3;
A22: x in (dom (Sq_Circ ")) /\ K0 by A20, RELAT_1:61;
then A23: x in K0 by XBOOLE_0:def_4;
then reconsider p = x as Point of (TOP-REAL 2) by A14;
 K00 = the carrier of ((TOP-REAL 2) | K00) by PRE_TOPC:8;
then  p in the carrier of ((TOP-REAL 2) | K00) by A22, XBOOLE_0:def_4;
then A24: p `1 <> 0 by A16;
set p9 = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]|;
A25: ( |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) & |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `2 = (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ) by EUCLID:52;
A26: ex px being Point of (TOP-REAL 2) st
( x = px & ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) by A23;
then A27: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by Th28;
A28: sqrt (1 + (((p `2) / (p `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
then  ( ( (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) & (- (p `1)) * (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ) or ( (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) >= (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) & (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (- (p `1)) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ) ) by A26, XREAL_1:64;
then A29: ( ( (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) <= (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) & - ((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))) <= (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) ) or ( (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) >= (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) & (p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))) <= - ((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))) ) ) ;
A30: |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| `1 = (p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52;
A31: now__::_thesis:_not_|[((p_`1)_*_(sqrt_(1_+_(((p_`2)_/_(p_`1))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`2)_/_(p_`1))_^2))))]|_=_0._(TOP-REAL_2)
assume  |[((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) * (sqrt (1 + (((p `2) / (p `1)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then  0 / (sqrt (1 + (((p `2) / (p `1)) ^2))) = ((p `1) * (sqrt (1 + (((p `2) / (p `1)) ^2)))) / (sqrt (1 + (((p `2) / (p `1)) ^2))) by A30, EUCLID:52, EUCLID:54;
hence  contradiction by A24, A28, XCMPLX_1:89; ::_thesis: verum
end;
 (Sq_Circ ") . p = y by A21, A23, FUNCT_1:49;
then  y in K0 by A31, A27, A29, A25;
hence  y in the carrier of (((TOP-REAL 2) | D) | K0) by PRE_TOPC:8; ::_thesis: verum
end;
dom ((Sq_Circ ") | K0) = (dom (Sq_Circ ")) /\ K0 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K0 by Th29, FUNCT_2:def_1
.= K0 by A14, XBOOLE_1:28 ;
then reconsider f = (Sq_Circ ") | K0 as Function of (((TOP-REAL 2) | D) | K0),((TOP-REAL 2) | D) by A7, A15, FUNCT_2:2, XBOOLE_1:1;
A32: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5;
A33: 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 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
hence  z in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
A34: rng ((Sq_Circ ") | K1) c= the carrier of (((TOP-REAL 2) | D) | K1)
proof
reconsider K10 = K1 as Subset of (TOP-REAL 2) by A33;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((Sq_Circ ") | K1) or y in the carrier of (((TOP-REAL 2) | D) | K1) )
A35: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K10) holds
q `2 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K10) implies q `2 <> 0 )
A36: the carrier of ((TOP-REAL 2) | K10) = K1 by PRE_TOPC:8;
assume  q in the carrier of ((TOP-REAL 2) | K10) ; ::_thesis: q `2 <> 0
then A37: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A36;
now__::_thesis:_not_q_`2_=_0
assume A38: q `2 = 0 ; ::_thesis: contradiction
then  q `1 = 0 by A37;
hence  contradiction by A37, A38, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence  q `2 <> 0 ; ::_thesis: verum
end;
assume  y in rng ((Sq_Circ ") | K1) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K1)
then consider x being set  such that
A39: x in dom ((Sq_Circ ") | K1) and
A40: y = ((Sq_Circ ") | K1) . x by FUNCT_1:def_3;
A41: x in (dom (Sq_Circ ")) /\ K1 by A39, RELAT_1:61;
then A42: x in K1 by XBOOLE_0:def_4;
then reconsider p = x as Point of (TOP-REAL 2) by A33;
 K10 = the carrier of ((TOP-REAL 2) | K10) by PRE_TOPC:8;
then  p in the carrier of ((TOP-REAL 2) | K10) by A41, XBOOLE_0:def_4;
then A43: p `2 <> 0 by A35;
set p9 = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]|;
A44: ( |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) & |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `1 = (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ) by EUCLID:52;
A45: ex px being Point of (TOP-REAL 2) st
( x = px & ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) by A42;
then A46: (Sq_Circ ") . p = |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by Th30;
A47: sqrt (1 + (((p `1) / (p `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
then  ( ( (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) & (- (p `2)) * (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ) or ( (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) >= (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) & (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (- (p `2)) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ) ) by A45, XREAL_1:64;
then A48: ( ( (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) <= (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) & - ((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2)))) <= (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) ) or ( (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) >= (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) & (p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2))) <= - ((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2)))) ) ) ;
A49: |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| `2 = (p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52;
A50: now__::_thesis:_not_|[((p_`1)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2)))),((p_`2)_*_(sqrt_(1_+_(((p_`1)_/_(p_`2))_^2))))]|_=_0._(TOP-REAL_2)
assume  |[((p `1) * (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then  0 / (sqrt (1 + (((p `1) / (p `2)) ^2))) = ((p `2) * (sqrt (1 + (((p `1) / (p `2)) ^2)))) / (sqrt (1 + (((p `1) / (p `2)) ^2))) by A49, EUCLID:52, EUCLID:54;
hence  contradiction by A43, A47, XCMPLX_1:89; ::_thesis: verum
end;
 (Sq_Circ ") . p = y by A40, A42, FUNCT_1:49;
then  y in K1 by A50, A46, A48, A44;
hence  y in the carrier of (((TOP-REAL 2) | D) | K1) by PRE_TOPC:8; ::_thesis: verum
end;
dom ((Sq_Circ ") | K1) = (dom (Sq_Circ ")) /\ K1 by RELAT_1:61
.= the carrier of (TOP-REAL 2) /\ K1 by Th29, FUNCT_2:def_1
.= K1 by A33, XBOOLE_1:28 ;
then reconsider g = (Sq_Circ ") | K1 as Function of (((TOP-REAL 2) | D) | K1),((TOP-REAL 2) | D) by A10, A34, FUNCT_2:2, XBOOLE_1:1;
A51: dom g = K1 by A10, FUNCT_2:def_1;
 g = (Sq_Circ ") | K1 ;
then A52: K1 is closed by A4, Th40;
A53: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5;
A54: now__::_thesis:_for_x_being_set_st_x_in_([#]_(((TOP-REAL_2)_|_D)_|_K0))_/\_([#]_(((TOP-REAL_2)_|_D)_|_K1))_holds_
f_._x_=_g_._x
let x be set ; ::_thesis: ( x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) implies f . x = g . x )
assume A55: x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) ; ::_thesis: f . x = g . x
then  x in K0 by A53, XBOOLE_0:def_4;
then  f . x = (Sq_Circ ") . x by FUNCT_1:49;
hence  f . x = g . x by A32, A55, FUNCT_1:49; ::_thesis: verum
end;
 f = (Sq_Circ ") | K0 ;
then A56: K0 is closed by A4, Th39;
A57: dom f = K0 by A7, FUNCT_2:def_1;
 D = [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5;
then A58: ([#] (((TOP-REAL 2) | D) | K0)) \/ ([#] (((TOP-REAL 2) | D) | K1)) = [#] ((TOP-REAL 2) | D) by A53, A32, A11, XBOOLE_0:def_10;
A59: ( f is continuous & g is continuous ) by A4, Th39, Th40;
then consider h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) such that
A60: h = f +* g and
 h is continuous by A53, A32, A58, A56, A52, A54, JGRAPH_2:1;
 ( K0 = [#] (((TOP-REAL 2) | D) | K0) & K1 = [#] (((TOP-REAL 2) | D) | K1) ) by PRE_TOPC:def_5;
then A61: f tolerates g by A54, A57, A51, PARTFUN1:def_4;
A62: for x being set st x in dom h holds
h . x = ((Sq_Circ ") | D) . x
proof
let x be set ; ::_thesis: ( x in dom h implies h . x = ((Sq_Circ ") | D) . x )
assume A63: x in dom h ; ::_thesis: h . x = ((Sq_Circ ") | D) . x
then reconsider p = x as Point of (TOP-REAL 2) by A13, XBOOLE_0:def_5;
 not x in {(0. (TOP-REAL 2))} by A13, A63, XBOOLE_0:def_5;
then A64: x <> 0. (TOP-REAL 2) by TARSKI:def_1;
 x in the carrier of (TOP-REAL 2) \ (D `) by A3, A13, A63;
then A65: x in (D `) ` by SUBSET_1:def_4;
percases ( x in K0 or not x in K0 ) ;
supposeA66: x in K0 ; ::_thesis: h . x = ((Sq_Circ ") | D) . x
A67: ((Sq_Circ ") | D) . p = (Sq_Circ ") . p by A65, FUNCT_1:49
.= f . p by A66, FUNCT_1:49 ;
h . p = (g +* f) . p by A60, A61, FUNCT_4:34
.= f . p by A57, A66, FUNCT_4:13 ;
hence  h . x = ((Sq_Circ ") | D) . x by A67; ::_thesis: verum
end;
suppose not x in K0 ; ::_thesis: h . x = ((Sq_Circ ") | D) . x
then  ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A64;
then  ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by XREAL_1:26;
then A68: x in K1 by A64;
((Sq_Circ ") | D) . p = (Sq_Circ ") . p by A65, FUNCT_1:49
.= g . p by A68, FUNCT_1:49 ;
hence  h . x = ((Sq_Circ ") | D) . x by A60, A51, A68, 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 = (Sq_Circ ") | D by A60, A2, A62, FUNCT_1:2;
hence  ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = (Sq_Circ ") | D & h is continuous ) by A53, A32, A58, A56, A59, A52, A54, JGRAPH_2:1; ::_thesis: verum
end;

theorem Th42: :: JGRAPH_3:42
ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h = Sq_Circ " & h is continuous )
proof
reconsider f = Sq_Circ " as Function of (TOP-REAL 2),(TOP-REAL 2) by Th29;
reconsider D = NonZero (TOP-REAL 2) as non empty Subset of (TOP-REAL 2) by JGRAPH_2:9;
A1: f . (0. (TOP-REAL 2)) = 0. (TOP-REAL 2) by Th28;
A2: 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))
A3: [#] ((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 A3, XBOOLE_0:def_5;
then A4: not p = 0. (TOP-REAL 2) by TARSKI:def_1;
percases ( ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) or ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ;
supposeA5: ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2))
then A6: q `2 <> 0 ;
set q9 = |[((q `1) * (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) * (sqrt (1 + (((q `1) / (q `2)) ^2))))]|;
A7: |[((q `1) * (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) * (sqrt (1 + (((q `1) / (q `2)) ^2))))]| `2 = (q `2) * (sqrt (1 + (((q `1) / (q `2)) ^2))) by EUCLID:52;
A8: sqrt (1 + (((q `1) / (q `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
now__::_thesis:_not_|[((q_`1)_*_(sqrt_(1_+_(((q_`1)_/_(q_`2))_^2)))),((q_`2)_*_(sqrt_(1_+_(((q_`1)_/_(q_`2))_^2))))]|_=_0._(TOP-REAL_2)
assume  |[((q `1) * (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) * (sqrt (1 + (((q `1) / (q `2)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then  0 * (q `2) = (q `2) * (sqrt (1 + (((q `1) / (q `2)) ^2))) by A7, EUCLID:52, EUCLID:54;
then  0 * (sqrt (1 + (((q `1) / (q `2)) ^2))) = ((q `2) * (sqrt (1 + (((q `1) / (q `2)) ^2)))) / (sqrt (1 + (((q `1) / (q `2)) ^2))) ;
hence  contradiction by A6, A8, XCMPLX_1:89; ::_thesis: verum
end;
hence  f . p <> f . (0. (TOP-REAL 2)) by A1, A5, Th28; ::_thesis: verum
end;
supposeA9: ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: f . p <> f . (0. (TOP-REAL 2))
A10: now__::_thesis:_not_q_`1_=_0
assume A11: q `1 = 0 ; ::_thesis: contradiction
then  q `2 = 0 by A9;
hence  contradiction by A4, A11, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
set q9 = |[((q `1) * (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) * (sqrt (1 + (((q `2) / (q `1)) ^2))))]|;
A12: |[((q `1) * (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) * (sqrt (1 + (((q `2) / (q `1)) ^2))))]| `1 = (q `1) * (sqrt (1 + (((q `2) / (q `1)) ^2))) by EUCLID:52;
A13: sqrt (1 + (((q `2) / (q `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
now__::_thesis:_not_|[((q_`1)_*_(sqrt_(1_+_(((q_`2)_/_(q_`1))_^2)))),((q_`2)_*_(sqrt_(1_+_(((q_`2)_/_(q_`1))_^2))))]|_=_0._(TOP-REAL_2)
assume  |[((q `1) * (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) * (sqrt (1 + (((q `2) / (q `1)) ^2))))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then  0 / (sqrt (1 + (((q `2) / (q `1)) ^2))) = ((q `1) * (sqrt (1 + (((q `2) / (q `1)) ^2)))) / (sqrt (1 + (((q `2) / (q `1)) ^2))) by A12, EUCLID:52, EUCLID:54;
hence  contradiction by A10, A13, XCMPLX_1:89; ::_thesis: verum
end;
hence  f . p <> f . (0. (TOP-REAL 2)) by A1, A4, A9, Th28; ::_thesis: verum
end;
end;
end;
A14: 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 Lm16;
assume that
A15: f . (0. (TOP-REAL 2)) in V and
A16: 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 A16, Lm16, PRE_TOPC:30;
then consider r being real number  such that
A17: r > 0 and
A18: Ball (u0,r) c= V by A1, A15, TOPMETR:15;
reconsider r = r as Real by XREAL_0:def_1;
reconsider W1 = Ball (u0,r), V1 = Ball (u0,(r / (sqrt 2))) as Subset of (TOP-REAL 2) by EUCLID:67;
A19: f .: V1 c= W1
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f .: V1 or z in W1 )
A20: sqrt 2 > 0 by SQUARE_1:25;
assume  z in f .: V1 ; ::_thesis: z in W1
then consider y being set  such that
A21: y in dom f and
A22: y in V1 and
A23: z = f . y by FUNCT_1:def_6;
 z in rng f by A21, A23, 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 A21;
reconsider qy = q as Point of (Euclid 2) by EUCLID:67;
A24: (q `1) ^2 >= 0 by XREAL_1:63;
A25: (q `2) ^2 >= 0 by XREAL_1:63;
 dist (u0,qy) < r / (sqrt 2) by A22, METRIC_1:11;
then  |.((0. (TOP-REAL 2)) - q).| < r / (sqrt 2) by JGRAPH_1:28;
then  sqrt (((((0. (TOP-REAL 2)) - q) `1) ^2) + ((((0. (TOP-REAL 2)) - q) `2) ^2)) < r / (sqrt 2) by JGRAPH_1:30;
then  sqrt (((((0. (TOP-REAL 2)) `1) - (q `1)) ^2) + ((((0. (TOP-REAL 2)) - q) `2) ^2)) < r / (sqrt 2) by TOPREAL3:3;
then  sqrt (((((0. (TOP-REAL 2)) `1) - (q `1)) ^2) + ((((0. (TOP-REAL 2)) `2) - (q `2)) ^2)) < r / (sqrt 2) by TOPREAL3:3;
then  (sqrt (((q `1) ^2) + ((q `2) ^2))) * (sqrt 2) < (r / (sqrt 2)) * (sqrt 2) by A20, JGRAPH_2:3, XREAL_1:68;
then  sqrt ((((q `1) ^2) + ((q `2) ^2)) * 2) < (r / (sqrt 2)) * (sqrt 2) by A24, A25, SQUARE_1:29;
then A26: sqrt ((((q `1) ^2) + ((q `2) ^2)) * 2) < r by A20, XCMPLX_1:87;
percases ( q = 0. (TOP-REAL 2) or ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) or ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ;
suppose q = 0. (TOP-REAL 2) ; ::_thesis: z in W1
then  z = 0. (TOP-REAL 2) by A23, Th28;
hence  z in W1 by A17, GOBOARD6:1; ::_thesis: verum
end;
supposeA27: ( q <> 0. (TOP-REAL 2) & ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ; ::_thesis: z in W1
A28: now__::_thesis:_not_(q_`1)_^2_<=_0
assume  (q `1) ^2 <= 0 ; ::_thesis: contradiction
then  (q `1) ^2 = 0 by XREAL_1:63;
then A29: q `1 = 0 by XCMPLX_1:6;
then  q `2 = 0 by A27;
hence  contradiction by A27, A29, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A30: (Sq_Circ ") . q = |[((q `1) * (sqrt (1 + (((q `2) / (q `1)) ^2)))),((q `2) * (sqrt (1 + (((q `2) / (q `1)) ^2))))]| by A27, Th28;
then  qz `1 = (q `1) * (sqrt (1 + (((q `2) / (q `1)) ^2))) by A23, EUCLID:52;
then A31: (qz `1) ^2 = ((q `1) ^2) * ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2) ;
 qz `2 = (q `2) * (sqrt (1 + (((q `2) / (q `1)) ^2))) by A23, A30, EUCLID:52;
then A32: (qz `2) ^2 = ((q `2) ^2) * ((sqrt (1 + (((q `2) / (q `1)) ^2))) ^2) ;
A33: 1 + (((q `2) / (q `1)) ^2) > 0 by Lm1;
now__::_thesis:_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_&_(q_`2)_^2_<=_(q_`1)_^2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_&_(q_`2)_^2_<=_(q_`1)_^2_)_)
percases ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) by A27;
caseA34: ( q `2 <= q `1 & - (q `1) <= q `2 ) ; ::_thesis: (q `2) ^2 <= (q `1) ^2
now__::_thesis:_(_(_0_<=_q_`2_&_(q_`2)_^2_<=_(q_`1)_^2_)_or_(_0_>_q_`2_&_(q_`2)_^2_<=_(q_`1)_^2_)_)
percases ( 0 <= q `2 or 0 > q `2 ) ;
case 0 <= q `2 ; ::_thesis: (q `2) ^2 <= (q `1) ^2
hence  (q `2) ^2 <= (q `1) ^2 by A34, SQUARE_1:15; ::_thesis: verum
end;
caseA35: 0 > q `2 ; ::_thesis: (q `2) ^2 <= (q `1) ^2
 - (- (q `1)) >= - (q `2) by A34, XREAL_1:24;
then  (- (q `2)) ^2 <= (q `1) ^2 by A35, SQUARE_1:15;
hence  (q `2) ^2 <= (q `1) ^2 ; ::_thesis: verum
end;
end;
end;
hence  (q `2) ^2 <= (q `1) ^2 ; ::_thesis: verum
end;
caseA36: ( q `2 >= q `1 & q `2 <= - (q `1) ) ; ::_thesis: (q `2) ^2 <= (q `1) ^2
now__::_thesis:_(_(_0_>=_q_`2_&_(q_`2)_^2_<=_(q_`1)_^2_)_or_(_0_<_q_`2_&_(q_`2)_^2_<=_(q_`1)_^2_)_)
percases ( 0 >= q `2 or 0 < q `2 ) ;
caseA37: 0 >= q `2 ; ::_thesis: (q `2) ^2 <= (q `1) ^2
 - (q `2) <= - (q `1) by A36, XREAL_1:24;
then  (- (q `2)) ^2 <= (- (q `1)) ^2 by A37, SQUARE_1:15;
hence  (q `2) ^2 <= (q `1) ^2 ; ::_thesis: verum
end;
case 0 < q `2 ; ::_thesis: (q `2) ^2 <= (q `1) ^2
then  (q `2) ^2 <= (- (q `1)) ^2 by A36, SQUARE_1:15;
hence  (q `2) ^2 <= (q `1) ^2 ; ::_thesis: verum
end;
end;
end;
hence  (q `2) ^2 <= (q `1) ^2 ; ::_thesis: verum
end;
end;
end;
then  ((q `2) ^2) / ((q `1) ^2) <= ((q `1) ^2) / ((q `1) ^2) by A28, XREAL_1:72;
then  ((q `2) / (q `1)) ^2 <= ((q `1) ^2) / ((q `1) ^2) by XCMPLX_1:76;
then  ((q `2) / (q `1)) ^2 <= 1 by A28, XCMPLX_1:60;
then A38: 1 + (((q `2) / (q `1)) ^2) <= 1 + 1 by XREAL_1:7;
then  ((q `2) ^2) * (1 + (((q `2) / (q `1)) ^2)) <= ((q `2) ^2) * 2 by A25, XREAL_1:64;
then A39: (qz `2) ^2 <= ((q `2) ^2) * 2 by A33, A32, SQUARE_1:def_2;
 ((q `1) ^2) * (1 + (((q `2) / (q `1)) ^2)) <= ((q `1) ^2) * 2 by A24, A38, XREAL_1:64;
then  (qz `1) ^2 <= ((q `1) ^2) * 2 by A33, A31, SQUARE_1:def_2;
then A40: ((qz `1) ^2) + ((qz `2) ^2) <= (((q `1) ^2) * 2) + (((q `2) ^2) * 2) by A39, XREAL_1:7;
 ( (qz `1) ^2 >= 0 & (qz `2) ^2 >= 0 ) by XREAL_1:63;
then A41: sqrt (((qz `1) ^2) + ((qz `2) ^2)) <= sqrt ((((q `1) ^2) * 2) + (((q `2) ^2) * 2)) by A40, SQUARE_1:26;
A42: ((0. (TOP-REAL 2)) - qz) `2 = ((0. (TOP-REAL 2)) `2) - (qz `2) by TOPREAL3:3
.= - (qz `2) by JGRAPH_2:3 ;
((0. (TOP-REAL 2)) - qz) `1 = ((0. (TOP-REAL 2)) `1) - (qz `1) by TOPREAL3:3
.= - (qz `1) by JGRAPH_2:3 ;
then  sqrt (((((0. (TOP-REAL 2)) - qz) `1) ^2) + ((((0. (TOP-REAL 2)) - qz) `2) ^2)) < r by A26, A42, A41, XXREAL_0:2;
then  |.((0. (TOP-REAL 2)) - qz).| < r by JGRAPH_1:30;
then  dist (u0,pz) < r by JGRAPH_1:28;
hence  z in W1 by METRIC_1:11; ::_thesis: verum
end;
supposeA43: ( q <> 0. (TOP-REAL 2) & not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: z in W1
A44: now__::_thesis:_not_(q_`2)_^2_<=_0
assume  (q `2) ^2 <= 0 ; ::_thesis: contradiction
then  (q `2) ^2 = 0 by XREAL_1:63;
then  q `2 = 0 by XCMPLX_1:6;
hence  contradiction by A43; ::_thesis: verum
end;
now__::_thesis:_(_(_q_`1_<=_q_`2_&_-_(q_`2)_<=_q_`1_&_(q_`1)_^2_<=_(q_`2)_^2_)_or_(_q_`1_>=_q_`2_&_q_`1_<=_-_(q_`2)_&_(q_`1)_^2_<=_(q_`2)_^2_)_)
percases ( ( q `1 <= q `2 & - (q `2) <= q `1 ) or ( q `1 >= q `2 & q `1 <= - (q `2) ) ) by A43, JGRAPH_2:13;
caseA45: ( q `1 <= q `2 & - (q `2) <= q `1 ) ; ::_thesis: (q `1) ^2 <= (q `2) ^2
now__::_thesis:_(_(_0_<=_q_`1_&_(q_`1)_^2_<=_(q_`2)_^2_)_or_(_0_>_q_`1_&_(q_`1)_^2_<=_(q_`2)_^2_)_)
percases ( 0 <= q `1 or 0 > q `1 ) ;
case 0 <= q `1 ; ::_thesis: (q `1) ^2 <= (q `2) ^2
hence  (q `1) ^2 <= (q `2) ^2 by A45, SQUARE_1:15; ::_thesis: verum
end;
caseA46: 0 > q `1 ; ::_thesis: (q `1) ^2 <= (q `2) ^2
 - (- (q `2)) >= - (q `1) by A45, XREAL_1:24;
then  (- (q `1)) ^2 <= (q `2) ^2 by A46, SQUARE_1:15;
hence  (q `1) ^2 <= (q `2) ^2 ; ::_thesis: verum
end;
end;
end;
hence  (q `1) ^2 <= (q `2) ^2 ; ::_thesis: verum
end;
caseA47: ( q `1 >= q `2 & q `1 <= - (q `2) ) ; ::_thesis: (q `1) ^2 <= (q `2) ^2
now__::_thesis:_(_(_0_>=_q_`1_&_(q_`1)_^2_<=_(q_`2)_^2_)_or_(_0_<_q_`1_&_(q_`1)_^2_<=_(q_`2)_^2_)_)
percases ( 0 >= q `1 or 0 < q `1 ) ;
caseA48: 0 >= q `1 ; ::_thesis: (q `1) ^2 <= (q `2) ^2
 - (q `1) <= - (q `2) by A47, XREAL_1:24;
then  (- (q `1)) ^2 <= (- (q `2)) ^2 by A48, SQUARE_1:15;
hence  (q `1) ^2 <= (q `2) ^2 ; ::_thesis: verum
end;
case 0 < q `1 ; ::_thesis: (q `1) ^2 <= (q `2) ^2
then  (q `1) ^2 <= (- (q `2)) ^2 by A47, SQUARE_1:15;
hence  (q `1) ^2 <= (q `2) ^2 ; ::_thesis: verum
end;
end;
end;
hence  (q `1) ^2 <= (q `2) ^2 ; ::_thesis: verum
end;
end;
end;
then  ((q `1) ^2) / ((q `2) ^2) <= ((q `2) ^2) / ((q `2) ^2) by A44, XREAL_1:72;
then  ((q `1) / (q `2)) ^2 <= ((q `2) ^2) / ((q `2) ^2) by XCMPLX_1:76;
then  ((q `1) / (q `2)) ^2 <= 1 by A44, XCMPLX_1:60;
then A49: 1 + (((q `1) / (q `2)) ^2) <= 1 + 1 by XREAL_1:7;
then A50: ((q `2) ^2) * (1 + (((q `1) / (q `2)) ^2)) <= ((q `2) ^2) * 2 by A25, XREAL_1:64;
 1 + (((q `1) / (q `2)) ^2) > 0 by Lm1;
then A51: (sqrt (1 + (((q `1) / (q `2)) ^2))) ^2 = 1 + (((q `1) / (q `2)) ^2) by SQUARE_1:def_2;
A52: ((q `1) ^2) * (1 + (((q `1) / (q `2)) ^2)) <= ((q `1) ^2) * 2 by A24, A49, XREAL_1:64;
A53: (Sq_Circ ") . q = |[((q `1) * (sqrt (1 + (((q `1) / (q `2)) ^2)))),((q `2) * (sqrt (1 + (((q `1) / (q `2)) ^2))))]| by A43, Th28;
then  qz `1 = (q `1) * (sqrt (1 + (((q `1) / (q `2)) ^2))) by A23, EUCLID:52;
then A54: (qz `1) ^2 <= ((q `1) ^2) * 2 by A52, A51, SQUARE_1:9;
 qz `2 = (q `2) * (sqrt (1 + (((q `1) / (q `2)) ^2))) by A23, A53, EUCLID:52;
then  (qz `2) ^2 <= ((q `2) ^2) * 2 by A50, A51, SQUARE_1:9;
then A55: ((qz `2) ^2) + ((qz `1) ^2) <= (((q `2) ^2) * 2) + (((q `1) ^2) * 2) by A54, XREAL_1:7;
 ( (qz `2) ^2 >= 0 & (qz `1) ^2 >= 0 ) by XREAL_1:63;
then A56: sqrt (((qz `2) ^2) + ((qz `1) ^2)) <= sqrt ((((q `2) ^2) * 2) + (((q `1) ^2) * 2)) by A55, SQUARE_1:26;
A57: ((0. (TOP-REAL 2)) - qz) `2 = ((0. (TOP-REAL 2)) `2) - (qz `2) by TOPREAL3:3
.= - (qz `2) by JGRAPH_2:3 ;
((0. (TOP-REAL 2)) - qz) `1 = ((0. (TOP-REAL 2)) `1) - (qz `1) by TOPREAL3:3
.= - (qz `1) by JGRAPH_2:3 ;
then  sqrt (((((0. (TOP-REAL 2)) - qz) `2) ^2) + ((((0. (TOP-REAL 2)) - qz) `1) ^2)) < r by A26, A57, A56, XXREAL_0:2;
then  |.((0. (TOP-REAL 2)) - qz).| < r by JGRAPH_1:30;
then  dist (u0,pz) < r by JGRAPH_1:28;
hence  z in W1 by METRIC_1:11; ::_thesis: verum
end;
end;
end;
A58: V1 is open by GOBOARD6:3;
 sqrt 2 > 0 by SQUARE_1:25;
then  u0 in V1 by A17, GOBOARD6:1, XREAL_1:139;
hence  ex W being Subset of (TOP-REAL 2) st
( 0. (TOP-REAL 2) in W & W is open & f .: W c= V ) by A18, A58, A19, XBOOLE_1:1; ::_thesis: verum
end;
A59: D ` = {(0. (TOP-REAL 2))} by Th20;
then  ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = (Sq_Circ ") | D & h is continuous ) by Th41;
hence  ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h = Sq_Circ " & h is continuous ) by A1, A59, A2, A14, Th3; ::_thesis: verum
end;

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 ) )
proof
thus  Sq_Circ is Function of (TOP-REAL 2),(TOP-REAL 2) ; ::_thesis: ( 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 ) )
A1: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds
( rng Sq_Circ = the carrier of (TOP-REAL 2) & f is being_homeomorphism )
proof
let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( f = Sq_Circ implies ( rng Sq_Circ = the carrier of (TOP-REAL 2) & f is being_homeomorphism ) )
assume A2: f = Sq_Circ ; ::_thesis: ( rng Sq_Circ = the carrier of (TOP-REAL 2) & f is being_homeomorphism )
reconsider g = f /" as Function of (TOP-REAL 2),(TOP-REAL 2) ;
A3: 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_=_0._(TOP-REAL_2)_&_ex_x_being_set_st_
(_x_in_dom_Sq_Circ_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_ex_x_being_set_st_
(_x_in_dom_Sq_Circ_&_y_=_Sq_Circ_._x_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_ex_x_being_set_st_
(_x_in_dom_Sq_Circ_&_y_=_Sq_Circ_._x_)_)_)
percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ;
case p2 = 0. (TOP-REAL 2) ; ::_thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )
then  y = Sq_Circ . p2 by Def1;
hence  ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A2, A3; ::_thesis: verum
end;
caseA4: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )
set px = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]|;
A5: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A6: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`1_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`2)_/_(p2_`1))_^2))))]|_`2_=_0_)
assume that
A7: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 and
A8: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = 0 ; ::_thesis: contradiction
 (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A8, EUCLID:52;
then A9: p2 `2 = 0 by A5, XCMPLX_1:6;
 (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A7, EUCLID:52;
then  p2 `1 = 0 by A5, XCMPLX_1:6;
hence  contradiction by A4, A9, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A10: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
A11: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52;
A12: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52;
then A13: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) = (p2 `2) / (p2 `1) by A11, A5, XCMPLX_1:91;
then A14: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `2 by A12, A5, XCMPLX_1:89;
 ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A4, A5, XREAL_1:64;
then  ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A11, A12, A5, XREAL_1:64;
then  ( ( (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A11, A5, EUCLID:52, XREAL_1:64;
then A15: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))))]| by A11, A12, A6, Def1, JGRAPH_2:3;
 (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `1 by A11, A5, A13, XCMPLX_1:89;
hence  ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A15, A14, A10, EUCLID:53; ::_thesis: verum
end;
caseA16: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )
set px = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]|;
A17: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A18: now__::_thesis:_(_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`2_=_0_implies_not_|[((p2_`1)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2)))),((p2_`2)_*_(sqrt_(1_+_(((p2_`1)_/_(p2_`2))_^2))))]|_`1_=_0_)
assume that
A19: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 and
 |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = 0 ; ::_thesis: contradiction
 (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A19, EUCLID:52;
then  p2 `2 = 0 by A17, XCMPLX_1:6;
hence  contradiction by A16; ::_thesis: verum
end;
A20: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52;
A21: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52;
then A22: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) = (p2 `1) / (p2 `2) by A20, A17, XCMPLX_1:91;
then A23: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `1 by A21, A17, XCMPLX_1:89;
 ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A16, JGRAPH_2:13;
then  ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A17, XREAL_1:64;
then  ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A20, A21, A17, XREAL_1:64;
then  ( ( (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A20, A17, EUCLID:52, XREAL_1:64;
then A24: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))))]| by A20, A21, A18, Th4, JGRAPH_2:3;
A25: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
 (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `2 by A20, A17, A22, XCMPLX_1:89;
hence  ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A24, A23, A25, EUCLID:53; ::_thesis: verum
end;
end;
end;
hence  y in rng f by A2, FUNCT_1:def_3; ::_thesis: verum
end;
then  rng f = the carrier of (TOP-REAL 2) by XBOOLE_0:def_10;
then A26: f is onto by FUNCT_2:def_3;
A27: rng f = dom (f ") by A2, FUNCT_1:33
.= dom (f /") by A2, A26, TOPS_2:def_4
.= [#] (TOP-REAL 2) by FUNCT_2:def_1 ;
 g = Sq_Circ " by A26, A2, TOPS_2:def_4;
hence  ( rng Sq_Circ = the carrier of (TOP-REAL 2) & f is being_homeomorphism ) by A2, A3, A27, Th21, Th42, TOPS_2:def_5; ::_thesis: verum
end;
hence  rng Sq_Circ = the carrier of (TOP-REAL 2) ; ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds
f is being_homeomorphism
thus  for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds
f is being_homeomorphism by A1; ::_thesis: verum
end;

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
proof
A1: dom (Sq_Circ ") = the carrier of (TOP-REAL 2) by Th29, FUNCT_2:def_1;
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 =  {  p where p is Point of (TOP-REAL 2) : |.p.| <= 1  }  & KXP =  {  q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) )  }  & KXN =  {  q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) )  }  & KYP =  {  q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) )  }  & KYN =  {  q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) )  }  & f . O in 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
let C0, KXP, KXN, KYP, KYN be Subset of (TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 =  {  p where p is Point of (TOP-REAL 2) : |.p.| <= 1  }  & KXP =  {  q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) )  }  & KXN =  {  q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) )  }  & KYP =  {  q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) )  }  & KYN =  {  q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) )  }  & f . O in 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
let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 =  {  p where p is Point of (TOP-REAL 2) : |.p.| <= 1  }  & KXP =  {  q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) )  }  & KXN =  {  q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) )  }  & KYP =  {  q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) )  }  & KYN =  {  q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) )  }  & f . O in KXN & f . I in KXP & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A2: ( 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 ) ; ::_thesis: rng f meets rng g
then consider p1 being Point of (TOP-REAL 2) such that
A3: f . O = p1 and
A4: |.p1.| = 1 and
A5: p1 `2 >= p1 `1 and
A6: p1 `2 <= - (p1 `1) ;
reconsider gg = (Sq_Circ ") * g as Function of I[01],(TOP-REAL 2) by Th29, FUNCT_2:13;
A7: dom g = the carrier of I[01] by FUNCT_2:def_1;
reconsider ff = (Sq_Circ ") * f as Function of I[01],(TOP-REAL 2) by Th29, FUNCT_2:13;
A8: dom gg = the carrier of I[01] by FUNCT_2:def_1;
A9: dom ff = the carrier of I[01] by FUNCT_2:def_1;
then A10: ff . O = (Sq_Circ ") . (f . O) by FUNCT_1:12;
A11: dom f = the carrier of I[01] by FUNCT_2:def_1;
A12: for r being Point of I[01] holds
( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 )
proof
let r be Point of I[01]; ::_thesis: ( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 )
 f . r in rng f by A11, FUNCT_1:3;
then  f . r in C0 by A2;
then consider p1 being Point of (TOP-REAL 2) such that
A13: f . r = p1 and
A14: |.p1.| <= 1 by A2;
 g . r in rng g by A7, FUNCT_1:3;
then  g . r in C0 by A2;
then consider p2 being Point of (TOP-REAL 2) such that
A15: g . r = p2 and
A16: |.p2.| <= 1 by A2;
A17: gg . r = (Sq_Circ ") . (g . r) by A8, FUNCT_1:12;
A18: now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_-_1_<=_(gg_._r)_`1_&_(gg_._r)_`1_<=_1_&_-_1_<=_(gg_._r)_`2_&_(gg_._r)_`2_<=_1_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_-_1_<=_(gg_._r)_`1_&_(gg_._r)_`1_<=_1_&_-_1_<=_(gg_._r)_`2_&_(gg_._r)_`2_<=_1_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_-_1_<=_(gg_._r)_`1_&_(gg_._r)_`1_<=_1_&_-_1_<=_(gg_._r)_`2_&_(gg_._r)_`2_<=_1_)_)
percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ;
case p2 = 0. (TOP-REAL 2) ; ::_thesis: ( - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 )
hence  ( - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 ) by A17, A15, Th28, JGRAPH_2:3; ::_thesis: verum
end;
caseA19: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: ( - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 )
set px = gg . r;
A20: (Sq_Circ ") . p2 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A19, Th28;
then A21: (gg . r) `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A17, A15, EUCLID:52;
 |.p2.| ^2 <= |.p2.| by A16, SQUARE_1:42;
then A22: |.p2.| ^2 <= 1 by A16, XXREAL_0:2;
A23: ((gg . r) `2) ^2 >= 0 by XREAL_1:63;
A24: ((gg . r) `1) ^2 >= 0 by XREAL_1:63;
A25: (gg . r) `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A17, A15, A20, EUCLID:52;
A26: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
then  ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A19, XREAL_1:64;
then A27: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( (gg . r) `2 >= (gg . r) `1 & (gg . r) `2 <= - ((gg . r) `1) ) ) by A21, A25, A26, XREAL_1:64;
then A28: ( ( (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) & - ((gg . r) `1) <= (gg . r) `2 ) or ( (gg . r) `2 >= (gg . r) `1 & (gg . r) `2 <= - ((gg . r) `1) ) ) by A17, A15, A20, A21, A26, EUCLID:52, XREAL_1:64;
A29: now__::_thesis:_(_(gg_._r)_`1_=_0_implies_not_(gg_._r)_`2_=_0_)
assume  ( (gg . r) `1 = 0 & (gg . r) `2 = 0 ) ; ::_thesis: contradiction
then  ( p2 `1 = 0 & p2 `2 = 0 ) by A21, A25, A26, XCMPLX_1:6;
hence  contradiction by A19, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
then A30: (gg . r) `1 <> 0 by A21, A25, A26, A27, XREAL_1:64;
set q = gg . r;
A31: |[(((gg . r) `1) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)))),(((gg . r) `2) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))))]| `2 = ((gg . r) `2) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) by EUCLID:52;
A32: 1 + ((((gg . r) `2) / ((gg . r) `1)) ^2) > 0 by Lm1;
A33: ( p2 = Sq_Circ . (gg . r) & |[(((gg . r) `1) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)))),(((gg . r) `2) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))))]| `1 = ((gg . r) `1) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) ) by A17, A15, Th43, EUCLID:52, FUNCT_1:32;
 Sq_Circ . (gg . r) = |[(((gg . r) `1) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)))),(((gg . r) `2) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))))]| by A21, A25, A29, A28, Def1, JGRAPH_2:3;
then |.p2.| ^2 = ((((gg . r) `1) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)))) ^2) + ((((gg . r) `2) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)))) ^2) by A33, A31, JGRAPH_1:29
.= ((((gg . r) `1) ^2) / ((sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) ^2)) + ((((gg . r) `2) / (sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)))) ^2) by XCMPLX_1:76
.= ((((gg . r) `1) ^2) / ((sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) ^2)) + ((((gg . r) `2) ^2) / ((sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) ^2)) by XCMPLX_1:76
.= ((((gg . r) `1) ^2) / (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) + ((((gg . r) `2) ^2) / ((sqrt (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) ^2)) by A32, SQUARE_1:def_2
.= ((((gg . r) `1) ^2) / (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) + ((((gg . r) `2) ^2) / (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) by A32, SQUARE_1:def_2
.= ((((gg . r) `1) ^2) + (((gg . r) `2) ^2)) / (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)) by XCMPLX_1:62 ;
then  (((((gg . r) `1) ^2) + (((gg . r) `2) ^2)) / (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2))) * (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)) <= 1 * (1 + ((((gg . r) `2) / ((gg . r) `1)) ^2)) by A32, A22, XREAL_1:64;
then  (((gg . r) `1) ^2) + (((gg . r) `2) ^2) <= 1 + ((((gg . r) `2) / ((gg . r) `1)) ^2) by A32, XCMPLX_1:87;
then  (((gg . r) `1) ^2) + (((gg . r) `2) ^2) <= 1 + ((((gg . r) `2) ^2) / (((gg . r) `1) ^2)) by XCMPLX_1:76;
then  ((((gg . r) `1) ^2) + (((gg . r) `2) ^2)) - 1 <= (((gg . r) `2) ^2) / (((gg . r) `1) ^2) by XREAL_1:20;
then  (((((gg . r) `1) ^2) + (((gg . r) `2) ^2)) - 1) * (((gg . r) `1) ^2) <= ((((gg . r) `2) ^2) / (((gg . r) `1) ^2)) * (((gg . r) `1) ^2) by A24, XREAL_1:64;
then  (((((gg . r) `1) ^2) + (((gg . r) `2) ^2)) - 1) * (((gg . r) `1) ^2) <= ((gg . r) `2) ^2 by A30, XCMPLX_1:6, XCMPLX_1:87;
then A34: ((((gg . r) `1) ^2) - 1) * ((((gg . r) `1) ^2) + (((gg . r) `2) ^2)) <= 0 by Lm19;
 (((gg . r) `1) ^2) + (((gg . r) `2) ^2) <> 0 by A29, COMPLEX1:1;
then A35: (((gg . r) `1) ^2) - 1 <= 0 by A24, A34, A23, XREAL_1:129;
then A36: (gg . r) `1 >= - 1 by SQUARE_1:43;
A37: (gg . r) `1 <= 1 by A35, SQUARE_1:43;
then  ( ( (gg . r) `2 <= 1 & - (- ((gg . r) `1)) >= - ((gg . r) `2) ) or ( (gg . r) `2 >= - 1 & (gg . r) `2 <= - ((gg . r) `1) ) ) by A21, A25, A28, A36, XREAL_1:24, XXREAL_0:2;
then  ( ( (gg . r) `2 <= 1 & (gg . r) `1 >= - ((gg . r) `2) ) or ( (gg . r) `2 >= - 1 & - ((gg . r) `2) >= - (- ((gg . r) `1)) ) ) by XREAL_1:24;
then  ( ( (gg . r) `2 <= 1 & 1 >= - ((gg . r) `2) ) or ( (gg . r) `2 >= - 1 & - ((gg . r) `2) >= (gg . r) `1 ) ) by A37, XXREAL_0:2;
then  ( ( (gg . r) `2 <= 1 & - 1 <= - (- ((gg . r) `2)) ) or ( (gg . r) `2 >= - 1 & - ((gg . r) `2) >= - 1 ) ) by A36, XREAL_1:24, XXREAL_0:2;
hence  ( - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 ) by A35, SQUARE_1:43, XREAL_1:24; ::_thesis: verum
end;
caseA38: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: ( - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 )
set pz = gg . r;
A39: (Sq_Circ ") . p2 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| by A38, Th28;
then A40: (gg . r) `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A17, A15, EUCLID:52;
A41: (gg . r) `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A17, A15, A39, EUCLID:52;
A42: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
 ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A38, JGRAPH_2:13;
then  ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A42, XREAL_1:64;
then A43: ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( (gg . r) `1 >= (gg . r) `2 & (gg . r) `1 <= - ((gg . r) `2) ) ) by A40, A41, A42, XREAL_1:64;
then A44: ( ( (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) & - ((gg . r) `2) <= (gg . r) `1 ) or ( (gg . r) `1 >= (gg . r) `2 & (gg . r) `1 <= - ((gg . r) `2) ) ) by A17, A15, A39, A40, A42, EUCLID:52, XREAL_1:64;
A45: now__::_thesis:_(_(gg_._r)_`2_=_0_implies_not_(gg_._r)_`1_=_0_)
assume that
A46: (gg . r) `2 = 0 and
 (gg . r) `1 = 0 ; ::_thesis: contradiction
 p2 `2 = 0 by A40, A42, A46, XCMPLX_1:6;
hence  contradiction by A38; ::_thesis: verum
end;
then A47: (gg . r) `2 <> 0 by A40, A41, A42, A43, XREAL_1:64;
A48: ( p2 = Sq_Circ . (gg . r) & |[(((gg . r) `1) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)))),(((gg . r) `2) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))))]| `2 = ((gg . r) `2) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) ) by A17, A15, Th43, EUCLID:52, FUNCT_1:32;
A49: ((gg . r) `2) ^2 >= 0 by XREAL_1:63;
 |.p2.| ^2 <= |.p2.| by A16, SQUARE_1:42;
then A50: |.p2.| ^2 <= 1 by A16, XXREAL_0:2;
A51: ((gg . r) `1) ^2 >= 0 by XREAL_1:63;
A52: |[(((gg . r) `1) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)))),(((gg . r) `2) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))))]| `1 = ((gg . r) `1) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) by EUCLID:52;
A53: 1 + ((((gg . r) `1) / ((gg . r) `2)) ^2) > 0 by Lm1;
 Sq_Circ . (gg . r) = |[(((gg . r) `1) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)))),(((gg . r) `2) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))))]| by A40, A41, A45, A44, Th4, JGRAPH_2:3;
then |.p2.| ^2 = ((((gg . r) `2) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)))) ^2) + ((((gg . r) `1) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)))) ^2) by A48, A52, JGRAPH_1:29
.= ((((gg . r) `2) ^2) / ((sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) ^2)) + ((((gg . r) `1) / (sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)))) ^2) by XCMPLX_1:76
.= ((((gg . r) `2) ^2) / ((sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) ^2)) + ((((gg . r) `1) ^2) / ((sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) ^2)) by XCMPLX_1:76
.= ((((gg . r) `2) ^2) / (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) + ((((gg . r) `1) ^2) / ((sqrt (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) ^2)) by A53, SQUARE_1:def_2
.= ((((gg . r) `2) ^2) / (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) + ((((gg . r) `1) ^2) / (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) by A53, SQUARE_1:def_2
.= ((((gg . r) `2) ^2) + (((gg . r) `1) ^2)) / (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)) by XCMPLX_1:62 ;
then  (((((gg . r) `2) ^2) + (((gg . r) `1) ^2)) / (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2))) * (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)) <= 1 * (1 + ((((gg . r) `1) / ((gg . r) `2)) ^2)) by A53, A50, XREAL_1:64;
then  (((gg . r) `2) ^2) + (((gg . r) `1) ^2) <= 1 + ((((gg . r) `1) / ((gg . r) `2)) ^2) by A53, XCMPLX_1:87;
then  (((gg . r) `2) ^2) + (((gg . r) `1) ^2) <= 1 + ((((gg . r) `1) ^2) / (((gg . r) `2) ^2)) by XCMPLX_1:76;
then  ((((gg . r) `2) ^2) + (((gg . r) `1) ^2)) - 1 <= (((gg . r) `1) ^2) / (((gg . r) `2) ^2) by XREAL_1:20;
then  (((((gg . r) `2) ^2) + (((gg . r) `1) ^2)) - 1) * (((gg . r) `2) ^2) <= ((((gg . r) `1) ^2) / (((gg . r) `2) ^2)) * (((gg . r) `2) ^2) by A49, XREAL_1:64;
then  (((((gg . r) `2) ^2) + (((gg . r) `1) ^2)) - 1) * (((gg . r) `2) ^2) <= ((gg . r) `1) ^2 by A47, XCMPLX_1:6, XCMPLX_1:87;
then A54: ((((gg . r) `2) ^2) - 1) * ((((gg . r) `2) ^2) + (((gg . r) `1) ^2)) <= 0 by Lm19;
 (((gg . r) `2) ^2) + (((gg . r) `1) ^2) <> 0 by A45, COMPLEX1:1;
then A55: (((gg . r) `2) ^2) - 1 <= 0 by A49, A54, A51, XREAL_1:129;
then A56: (gg . r) `2 >= - 1 by SQUARE_1:43;
A57: (gg . r) `2 <= 1 by A55, SQUARE_1:43;
then  ( ( (gg . r) `1 <= 1 & - (- ((gg . r) `2)) >= - ((gg . r) `1) ) or ( (gg . r) `1 >= - 1 & (gg . r) `1 <= - ((gg . r) `2) ) ) by A40, A41, A44, A56, XREAL_1:24, XXREAL_0:2;
then  ( ( (gg . r) `1 <= 1 & 1 >= - ((gg . r) `1) ) or ( (gg . r) `1 >= - 1 & - ((gg . r) `1) >= - (- ((gg . r) `2)) ) ) by A57, XREAL_1:24, XXREAL_0:2;
then  ( ( (gg . r) `1 <= 1 & 1 >= - ((gg . r) `1) ) or ( (gg . r) `1 >= - 1 & - ((gg . r) `1) >= - 1 ) ) by A56, XXREAL_0:2;
then  ( ( (gg . r) `1 <= 1 & - 1 <= - (- ((gg . r) `1)) ) or ( (gg . r) `1 >= - 1 & (gg . r) `1 <= 1 ) ) by XREAL_1:24;
hence  ( - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 ) by A55, SQUARE_1:43; ::_thesis: verum
end;
end;
end;
A58: ff . r = (Sq_Circ ") . (f . r) by A9, FUNCT_1:12;
now__::_thesis:_(_(_p1_=_0._(TOP-REAL_2)_&_-_1_<=_(ff_._r)_`1_&_(ff_._r)_`1_<=_1_&_-_1_<=_(ff_._r)_`2_&_(ff_._r)_`2_<=_1_)_or_(_p1_<>_0._(TOP-REAL_2)_&_(_(_p1_`2_<=_p1_`1_&_-_(p1_`1)_<=_p1_`2_)_or_(_p1_`2_>=_p1_`1_&_p1_`2_<=_-_(p1_`1)_)_)_&_-_1_<=_(ff_._r)_`1_&_(ff_._r)_`1_<=_1_&_-_1_<=_(ff_._r)_`2_&_(ff_._r)_`2_<=_1_)_or_(_p1_<>_0._(TOP-REAL_2)_&_not_(_p1_`2_<=_p1_`1_&_-_(p1_`1)_<=_p1_`2_)_&_not_(_p1_`2_>=_p1_`1_&_p1_`2_<=_-_(p1_`1)_)_&_-_1_<=_(ff_._r)_`1_&_(ff_._r)_`1_<=_1_&_-_1_<=_(ff_._r)_`2_&_(ff_._r)_`2_<=_1_)_)
percases ( p1 = 0. (TOP-REAL 2) or ( p1 <> 0. (TOP-REAL 2) & ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) or ( p1 <> 0. (TOP-REAL 2) & not ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) & not ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) ;
case p1 = 0. (TOP-REAL 2) ; ::_thesis: ( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 )
hence  ( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 ) by A58, A13, Th28, JGRAPH_2:3; ::_thesis: verum
end;
caseA59: ( p1 <> 0. (TOP-REAL 2) & ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) ; ::_thesis: ( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 )
set px = ff . r;
 (Sq_Circ ") . p1 = |[((p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| by A59, Th28;
then A60: ( (ff . r) `1 = (p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) & (ff . r) `2 = (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) by A58, A13, EUCLID:52;
A61: sqrt (1 + (((p1 `2) / (p1 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
then  ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) ) by A59, XREAL_1:64;
then A62: ( ( p1 `2 <= p1 `1 & (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) or ( (ff . r) `2 >= (ff . r) `1 & (ff . r) `2 <= - ((ff . r) `1) ) ) by A60, A61, XREAL_1:64;
then A63: ( ( (ff . r) `2 <= (ff . r) `1 & - ((ff . r) `1) <= (ff . r) `2 ) or ( (ff . r) `2 >= (ff . r) `1 & (ff . r) `2 <= - ((ff . r) `1) ) ) by A60, A61, XREAL_1:64;
A64: now__::_thesis:_(_(ff_._r)_`1_=_0_implies_not_(ff_._r)_`2_=_0_)
assume  ( (ff . r) `1 = 0 & (ff . r) `2 = 0 ) ; ::_thesis: contradiction
then  ( p1 `1 = 0 & p1 `2 = 0 ) by A60, A61, XCMPLX_1:6;
hence  contradiction by A59, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
then A65: (ff . r) `1 <> 0 by A60, A61, A62, XREAL_1:64;
 |.p1.| ^2 <= |.p1.| by A14, SQUARE_1:42;
then A66: |.p1.| ^2 <= 1 by A14, XXREAL_0:2;
A67: ((ff . r) `1) ^2 >= 0 by XREAL_1:63;
A68: ((ff . r) `2) ^2 >= 0 by XREAL_1:63;
set q = ff . r;
A69: |[(((ff . r) `1) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)))),(((ff . r) `2) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))))]| `2 = ((ff . r) `2) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) by EUCLID:52;
A70: 1 + ((((ff . r) `2) / ((ff . r) `1)) ^2) > 0 by Lm1;
A71: ( p1 = Sq_Circ . (ff . r) & |[(((ff . r) `1) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)))),(((ff . r) `2) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))))]| `1 = ((ff . r) `1) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) ) by A58, A13, Th43, EUCLID:52, FUNCT_1:32;
 Sq_Circ . (ff . r) = |[(((ff . r) `1) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)))),(((ff . r) `2) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))))]| by A64, A63, Def1, JGRAPH_2:3;
then |.p1.| ^2 = ((((ff . r) `1) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)))) ^2) + ((((ff . r) `2) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)))) ^2) by A71, A69, JGRAPH_1:29
.= ((((ff . r) `1) ^2) / ((sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) ^2)) + ((((ff . r) `2) / (sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)))) ^2) by XCMPLX_1:76
.= ((((ff . r) `1) ^2) / ((sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) ^2)) + ((((ff . r) `2) ^2) / ((sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) ^2)) by XCMPLX_1:76
.= ((((ff . r) `1) ^2) / (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) + ((((ff . r) `2) ^2) / ((sqrt (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) ^2)) by A70, SQUARE_1:def_2
.= ((((ff . r) `1) ^2) / (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) + ((((ff . r) `2) ^2) / (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) by A70, SQUARE_1:def_2
.= ((((ff . r) `1) ^2) + (((ff . r) `2) ^2)) / (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)) by XCMPLX_1:62 ;
then  (((((ff . r) `1) ^2) + (((ff . r) `2) ^2)) / (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2))) * (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)) <= 1 * (1 + ((((ff . r) `2) / ((ff . r) `1)) ^2)) by A70, A66, XREAL_1:64;
then  (((ff . r) `1) ^2) + (((ff . r) `2) ^2) <= 1 + ((((ff . r) `2) / ((ff . r) `1)) ^2) by A70, XCMPLX_1:87;
then  (((ff . r) `1) ^2) + (((ff . r) `2) ^2) <= 1 + ((((ff . r) `2) ^2) / (((ff . r) `1) ^2)) by XCMPLX_1:76;
then  ((((ff . r) `1) ^2) + (((ff . r) `2) ^2)) - 1 <= (((ff . r) `2) ^2) / (((ff . r) `1) ^2) by XREAL_1:20;
then  (((((ff . r) `1) ^2) + (((ff . r) `2) ^2)) - 1) * (((ff . r) `1) ^2) <= ((((ff . r) `2) ^2) / (((ff . r) `1) ^2)) * (((ff . r) `1) ^2) by A67, XREAL_1:64;
then  (((((ff . r) `1) ^2) + (((ff . r) `2) ^2)) - 1) * (((ff . r) `1) ^2) <= ((ff . r) `2) ^2 by A65, XCMPLX_1:6, XCMPLX_1:87;
then A72: ((((ff . r) `1) ^2) - 1) * ((((ff . r) `1) ^2) + (((ff . r) `2) ^2)) <= 0 by Lm19;
 (((ff . r) `1) ^2) + (((ff . r) `2) ^2) <> 0 by A64, COMPLEX1:1;
then A73: (((ff . r) `1) ^2) - 1 <= 0 by A67, A72, A68, XREAL_1:129;
then A74: (ff . r) `1 >= - 1 by SQUARE_1:43;
A75: (ff . r) `1 <= 1 by A73, SQUARE_1:43;
then  ( ( (ff . r) `2 <= 1 & - (- ((ff . r) `1)) >= - ((ff . r) `2) ) or ( (ff . r) `2 >= - 1 & (ff . r) `2 <= - ((ff . r) `1) ) ) by A63, A74, XREAL_1:24, XXREAL_0:2;
then  ( ( (ff . r) `2 <= 1 & (ff . r) `1 >= - ((ff . r) `2) ) or ( (ff . r) `2 >= - 1 & - ((ff . r) `2) >= - (- ((ff . r) `1)) ) ) by XREAL_1:24;
then  ( ( (ff . r) `2 <= 1 & 1 >= - ((ff . r) `2) ) or ( (ff . r) `2 >= - 1 & - ((ff . r) `2) >= (ff . r) `1 ) ) by A75, XXREAL_0:2;
then  ( ( (ff . r) `2 <= 1 & - 1 <= - (- ((ff . r) `2)) ) or ( (ff . r) `2 >= - 1 & - ((ff . r) `2) >= - 1 ) ) by A74, XREAL_1:24, XXREAL_0:2;
hence  ( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 ) by A73, SQUARE_1:43, XREAL_1:24; ::_thesis: verum
end;
caseA76: ( p1 <> 0. (TOP-REAL 2) & not ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) & not ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ; ::_thesis: ( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 )
set pz = ff . r;
A77: (Sq_Circ ") . p1 = |[((p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))),((p1 `2) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))))]| by A76, Th28;
then A78: (ff . r) `2 = (p1 `2) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A58, A13, EUCLID:52;
A79: (ff . r) `1 = (p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A58, A13, A77, EUCLID:52;
A80: sqrt (1 + (((p1 `1) / (p1 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
 ( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ) by A76, JGRAPH_2:13;
then  ( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & (p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) <= (- (p1 `2)) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ) ) by A80, XREAL_1:64;
then A81: ( ( p1 `1 <= p1 `2 & (- (p1 `2)) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) <= (p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ) or ( (ff . r) `1 >= (ff . r) `2 & (ff . r) `1 <= - ((ff . r) `2) ) ) by A78, A79, A80, XREAL_1:64;
then A82: ( ( (p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) <= (p1 `2) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) & - ((ff . r) `2) <= (ff . r) `1 ) or ( (ff . r) `1 >= (ff . r) `2 & (ff . r) `1 <= - ((ff . r) `2) ) ) by A58, A13, A77, A78, A80, EUCLID:52, XREAL_1:64;
A83: now__::_thesis:_(_(ff_._r)_`2_=_0_implies_not_(ff_._r)_`1_=_0_)
assume that
A84: (ff . r) `2 = 0 and
 (ff . r) `1 = 0 ; ::_thesis: contradiction
 p1 `2 = 0 by A78, A80, A84, XCMPLX_1:6;
hence  contradiction by A76; ::_thesis: verum
end;
then A85: (ff . r) `2 <> 0 by A78, A79, A80, A81, XREAL_1:64;
A86: ( p1 = Sq_Circ . (ff . r) & |[(((ff . r) `1) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)))),(((ff . r) `2) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))))]| `2 = ((ff . r) `2) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) ) by A58, A13, Th43, EUCLID:52, FUNCT_1:32;
A87: ((ff . r) `2) ^2 >= 0 by XREAL_1:63;
 |.p1.| ^2 <= |.p1.| by A14, SQUARE_1:42;
then A88: |.p1.| ^2 <= 1 by A14, XXREAL_0:2;
A89: ((ff . r) `1) ^2 >= 0 by XREAL_1:63;
A90: |[(((ff . r) `1) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)))),(((ff . r) `2) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))))]| `1 = ((ff . r) `1) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) by EUCLID:52;
A91: 1 + ((((ff . r) `1) / ((ff . r) `2)) ^2) > 0 by Lm1;
 Sq_Circ . (ff . r) = |[(((ff . r) `1) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)))),(((ff . r) `2) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))))]| by A78, A79, A83, A82, Th4, JGRAPH_2:3;
then |.p1.| ^2 = ((((ff . r) `2) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)))) ^2) + ((((ff . r) `1) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)))) ^2) by A86, A90, JGRAPH_1:29
.= ((((ff . r) `2) ^2) / ((sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) ^2)) + ((((ff . r) `1) / (sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)))) ^2) by XCMPLX_1:76
.= ((((ff . r) `2) ^2) / ((sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) ^2)) + ((((ff . r) `1) ^2) / ((sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) ^2)) by XCMPLX_1:76
.= ((((ff . r) `2) ^2) / (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) + ((((ff . r) `1) ^2) / ((sqrt (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) ^2)) by A91, SQUARE_1:def_2
.= ((((ff . r) `2) ^2) / (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) + ((((ff . r) `1) ^2) / (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) by A91, SQUARE_1:def_2
.= ((((ff . r) `2) ^2) + (((ff . r) `1) ^2)) / (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)) by XCMPLX_1:62 ;
then  (((((ff . r) `2) ^2) + (((ff . r) `1) ^2)) / (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2))) * (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)) <= 1 * (1 + ((((ff . r) `1) / ((ff . r) `2)) ^2)) by A91, A88, XREAL_1:64;
then  (((ff . r) `2) ^2) + (((ff . r) `1) ^2) <= 1 + ((((ff . r) `1) / ((ff . r) `2)) ^2) by A91, XCMPLX_1:87;
then  (((ff . r) `2) ^2) + (((ff . r) `1) ^2) <= 1 + ((((ff . r) `1) ^2) / (((ff . r) `2) ^2)) by XCMPLX_1:76;
then  ((((ff . r) `2) ^2) + (((ff . r) `1) ^2)) - 1 <= (((ff . r) `1) ^2) / (((ff . r) `2) ^2) by XREAL_1:20;
then  (((((ff . r) `2) ^2) + (((ff . r) `1) ^2)) - 1) * (((ff . r) `2) ^2) <= ((((ff . r) `1) ^2) / (((ff . r) `2) ^2)) * (((ff . r) `2) ^2) by A87, XREAL_1:64;
then  (((((ff . r) `2) ^2) + (((ff . r) `1) ^2)) - 1) * (((ff . r) `2) ^2) <= ((ff . r) `1) ^2 by A85, XCMPLX_1:6, XCMPLX_1:87;
then A92: ((((ff . r) `2) ^2) - 1) * ((((ff . r) `2) ^2) + (((ff . r) `1) ^2)) <= 0 by Lm19;
 (((ff . r) `2) ^2) + (((ff . r) `1) ^2) <> 0 by A83, COMPLEX1:1;
then A93: (((ff . r) `2) ^2) - 1 <= 0 by A87, A92, A89, XREAL_1:129;
then A94: (ff . r) `2 >= - 1 by SQUARE_1:43;
A95: (ff . r) `2 <= 1 by A93, SQUARE_1:43;
then  ( ( (ff . r) `1 <= 1 & - (- ((ff . r) `2)) >= - ((ff . r) `1) ) or ( (ff . r) `1 >= - 1 & (ff . r) `1 <= - ((ff . r) `2) ) ) by A78, A79, A82, A94, XREAL_1:24, XXREAL_0:2;
then  ( ( (ff . r) `1 <= 1 & 1 >= - ((ff . r) `1) ) or ( (ff . r) `1 >= - 1 & - ((ff . r) `1) >= - (- ((ff . r) `2)) ) ) by A95, XREAL_1:24, XXREAL_0:2;
then  ( ( (ff . r) `1 <= 1 & 1 >= - ((ff . r) `1) ) or ( (ff . r) `1 >= - 1 & - ((ff . r) `1) >= - 1 ) ) by A94, XXREAL_0:2;
then  ( ( (ff . r) `1 <= 1 & - 1 <= - (- ((ff . r) `1)) ) or ( (ff . r) `1 >= - 1 & (ff . r) `1 <= 1 ) ) by XREAL_1:24;
hence  ( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 ) by A93, SQUARE_1:43; ::_thesis: verum
end;
end;
end;
hence  ( - 1 <= (ff . r) `1 & (ff . r) `1 <= 1 & - 1 <= (gg . r) `1 & (gg . r) `1 <= 1 & - 1 <= (ff . r) `2 & (ff . r) `2 <= 1 & - 1 <= (gg . r) `2 & (gg . r) `2 <= 1 ) by A18; ::_thesis: verum
end;
set y = the Element of (rng ff) /\ (rng gg);
A96: p1 <> 0. (TOP-REAL 2) by A4, TOPRNS_1:23;
then A97: (Sq_Circ ") . p1 = |[((p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| by A5, A6, Th28;
 ( (ff . O) `1 = - 1 & (ff . I) `1 = 1 & (gg . O) `2 = - 1 & (gg . I) `2 = 1 )
proof
set pz = gg . O;
set py = ff . I;
set px = ff . O;
set q = ff . O;
A98: |[(((ff . O) `1) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)))),(((ff . O) `2) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))))]| `1 = ((ff . O) `1) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) by EUCLID:52;
set pu = gg . I;
A99: |[(((ff . I) `1) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)))),(((ff . I) `2) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))))]| `1 = ((ff . I) `1) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) by EUCLID:52;
A100: |[(((gg . I) `1) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)))),(((gg . I) `2) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))))]| `2 = ((gg . I) `2) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) by EUCLID:52;
A101: 1 + ((((gg . I) `1) / ((gg . I) `2)) ^2) > 0 by Lm1;
 (Sq_Circ ") . p1 = ff . O by A9, A3, FUNCT_1:12;
then A102: p1 = Sq_Circ . (ff . O) by Th43, FUNCT_1:32;
consider p4 being Point of (TOP-REAL 2) such that
A103: g . I = p4 and
A104: |.p4.| = 1 and
A105: p4 `2 >= p4 `1 and
A106: p4 `2 >= - (p4 `1) by A2;
A107: sqrt (1 + (((p4 `1) / (p4 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A108: - (p4 `2) <= - (- (p4 `1)) by A106, XREAL_1:24;
then A109: ( ( p4 `1 <= p4 `2 & (- (p4 `2)) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) <= (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) ) or ( (gg . I) `1 >= (gg . I) `2 & (gg . I) `1 <= - ((gg . I) `2) ) ) by A105, A107, XREAL_1:64;
A110: gg . I = (Sq_Circ ") . (g . I) by A8, FUNCT_1:12;
then A111: p4 = Sq_Circ . (gg . I) by A103, Th43, FUNCT_1:32;
A112: p4 <> 0. (TOP-REAL 2) by A104, TOPRNS_1:23;
then A113: (Sq_Circ ") . p4 = |[((p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2)))),((p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))))]| by A105, A108, Th30;
then A114: (gg . I) `2 = (p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) by A110, A103, EUCLID:52;
A115: (gg . I) `1 = (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) by A110, A103, A113, EUCLID:52;
A116: now__::_thesis:_(_(gg_._I)_`2_=_0_implies_not_(gg_._I)_`1_=_0_)
assume  ( (gg . I) `2 = 0 & (gg . I) `1 = 0 ) ; ::_thesis: contradiction
then  ( p4 `2 = 0 & p4 `1 = 0 ) by A114, A115, A107, XCMPLX_1:6;
hence  contradiction by A112, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
 ( ( (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) <= (p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) & - ((gg . I) `2) <= (gg . I) `1 ) or ( (gg . I) `1 >= (gg . I) `2 & (gg . I) `1 <= - ((gg . I) `2) ) ) by A110, A103, A113, A114, A107, A109, EUCLID:52, XREAL_1:64;
then A117: Sq_Circ . (gg . I) = |[(((gg . I) `1) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)))),(((gg . I) `2) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))))]| by A114, A115, A116, Th4, JGRAPH_2:3;
 |[(((gg . I) `1) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)))),(((gg . I) `2) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))))]| `1 = ((gg . I) `1) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) by EUCLID:52;
then |.p4.| ^2 = ((((gg . I) `2) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)))) ^2) + ((((gg . I) `1) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)))) ^2) by A111, A117, A100, JGRAPH_1:29
.= ((((gg . I) `2) ^2) / ((sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) ^2)) + ((((gg . I) `1) / (sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)))) ^2) by XCMPLX_1:76
.= ((((gg . I) `2) ^2) / ((sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) ^2)) + ((((gg . I) `1) ^2) / ((sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) ^2)) by XCMPLX_1:76
.= ((((gg . I) `2) ^2) / (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) + ((((gg . I) `1) ^2) / ((sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) ^2)) by A101, SQUARE_1:def_2
.= ((((gg . I) `2) ^2) / (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) + ((((gg . I) `1) ^2) / (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) by A101, SQUARE_1:def_2
.= ((((gg . I) `2) ^2) + (((gg . I) `1) ^2)) / (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)) by XCMPLX_1:62 ;
then  (((((gg . I) `2) ^2) + (((gg . I) `1) ^2)) / (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2))) * (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)) = 1 * (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)) by A104;
then  (((gg . I) `2) ^2) + (((gg . I) `1) ^2) = 1 + ((((gg . I) `1) / ((gg . I) `2)) ^2) by A101, XCMPLX_1:87;
then A118: ((((gg . I) `2) ^2) + (((gg . I) `1) ^2)) - 1 = (((gg . I) `1) ^2) / (((gg . I) `2) ^2) by XCMPLX_1:76;
 (gg . I) `2 <> 0 by A114, A115, A107, A116, A109, XREAL_1:64;
then  (((((gg . I) `2) ^2) + (((gg . I) `1) ^2)) - 1) * (((gg . I) `2) ^2) = ((gg . I) `1) ^2 by A118, XCMPLX_1:6, XCMPLX_1:87;
then A119: ((((gg . I) `2) ^2) - 1) * ((((gg . I) `2) ^2) + (((gg . I) `1) ^2)) = 0 ;
 (((gg . I) `2) ^2) + (((gg . I) `1) ^2) <> 0 by A116, COMPLEX1:1;
then A120: (((gg . I) `2) ^2) - 1 = 0 by A119, XCMPLX_1:6;
A121: sqrt (1 + (((p1 `2) / (p1 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A122: sqrt (1 + ((((gg . I) `1) / ((gg . I) `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A123: now__::_thesis:_not_(gg_._I)_`2_=_-_1
assume A124: (gg . I) `2 = - 1 ; ::_thesis: contradiction
then  - (p4 `1) < 0 by A106, A111, A117, A100, A122, XREAL_1:141;
then  - (- (p4 `1)) > - 0 ;
hence  contradiction by A105, A111, A117, A122, A124, EUCLID:52; ::_thesis: verum
end;
A125: 1 + ((((gg . O) `1) / ((gg . O) `2)) ^2) > 0 by Lm1;
A126: ( (ff . O) `1 = (p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) & (ff . O) `2 = (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) by A10, A3, A97, EUCLID:52;
A127: now__::_thesis:_(_(ff_._O)_`1_=_0_implies_not_(ff_._O)_`2_=_0_)
assume  ( (ff . O) `1 = 0 & (ff . O) `2 = 0 ) ; ::_thesis: contradiction
then  ( p1 `1 = 0 & p1 `2 = 0 ) by A126, A121, XCMPLX_1:6;
hence  contradiction by A96, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
 ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) ) by A5, A6, A121, XREAL_1:64;
then A128: ( ( p1 `2 <= p1 `1 & (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) or ( (ff . O) `2 >= (ff . O) `1 & (ff . O) `2 <= - ((ff . O) `1) ) ) by A126, A121, XREAL_1:64;
then  ( ( (ff . O) `2 <= (ff . O) `1 & - ((ff . O) `1) <= (ff . O) `2 ) or ( (ff . O) `2 >= (ff . O) `1 & (ff . O) `2 <= - ((ff . O) `1) ) ) by A126, A121, XREAL_1:64;
then A129: Sq_Circ . (ff . O) = |[(((ff . O) `1) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)))),(((ff . O) `2) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))))]| by A127, Def1, JGRAPH_2:3;
A130: sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A131: now__::_thesis:_not_(ff_._O)_`1_=_1
assume A132: (ff . O) `1 = 1 ; ::_thesis: contradiction
 - (p1 `2) >= - (- (p1 `1)) by A6, XREAL_1:24;
then  - (p1 `2) > 0 by A102, A129, A98, A130, A132, XREAL_1:139;
then  - (- (p1 `2)) < - 0 ;
hence  contradiction by A5, A102, A129, A130, A132, EUCLID:52; ::_thesis: verum
end;
consider p2 being Point of (TOP-REAL 2) such that
A133: f . I = p2 and
A134: |.p2.| = 1 and
A135: p2 `2 <= p2 `1 and
A136: p2 `2 >= - (p2 `1) by A2;
A137: ff . I = (Sq_Circ ") . (f . I) by A9, FUNCT_1:12;
then A138: p2 = Sq_Circ . (ff . I) by A133, Th43, FUNCT_1:32;
A139: p2 <> 0. (TOP-REAL 2) by A134, TOPRNS_1:23;
then A140: (Sq_Circ ") . p2 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A135, A136, Th28;
then A141: (ff . I) `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A137, A133, EUCLID:52;
A142: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A143: (ff . I) `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A137, A133, A140, EUCLID:52;
A144: now__::_thesis:_(_(ff_._I)_`1_=_0_implies_not_(ff_._I)_`2_=_0_)
assume  ( (ff . I) `1 = 0 & (ff . I) `2 = 0 ) ; ::_thesis: contradiction
then  ( p2 `1 = 0 & p2 `2 = 0 ) by A141, A143, A142, XCMPLX_1:6;
hence  contradiction by A139, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A145: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( (ff . I) `2 >= (ff . I) `1 & (ff . I) `2 <= - ((ff . I) `1) ) ) by A135, A136, A142, XREAL_1:64;
then  ( ( (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) & - ((ff . I) `1) <= (ff . I) `2 ) or ( (ff . I) `2 >= (ff . I) `1 & (ff . I) `2 <= - ((ff . I) `1) ) ) by A137, A133, A140, A141, A142, EUCLID:52, XREAL_1:64;
then A146: Sq_Circ . (ff . I) = |[(((ff . I) `1) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)))),(((ff . I) `2) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))))]| by A141, A143, A144, Def1, JGRAPH_2:3;
A147: sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)) > 0 by Lm1, SQUARE_1:25;
A148: now__::_thesis:_not_(ff_._I)_`1_=_-_1
assume A149: (ff . I) `1 = - 1 ; ::_thesis: contradiction
 - (p2 `2) <= - (- (p2 `1)) by A136, XREAL_1:24;
then  - (p2 `2) < 0 by A138, A146, A99, A147, A149, XREAL_1:141;
then  - (- (p2 `2)) > - 0 ;
hence  contradiction by A135, A138, A146, A147, A149, EUCLID:52; ::_thesis: verum
end;
A150: 1 + ((((ff . I) `2) / ((ff . I) `1)) ^2) > 0 by Lm1;
 |[(((ff . I) `1) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)))),(((ff . I) `2) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))))]| `2 = ((ff . I) `2) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) by EUCLID:52;
then |.p2.| ^2 = ((((ff . I) `1) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)))) ^2) + ((((ff . I) `2) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)))) ^2) by A138, A146, A99, JGRAPH_1:29
.= ((((ff . I) `1) ^2) / ((sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) ^2)) + ((((ff . I) `2) / (sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)))) ^2) by XCMPLX_1:76
.= ((((ff . I) `1) ^2) / ((sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) ^2)) + ((((ff . I) `2) ^2) / ((sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) ^2)) by XCMPLX_1:76
.= ((((ff . I) `1) ^2) / (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) + ((((ff . I) `2) ^2) / ((sqrt (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) ^2)) by A150, SQUARE_1:def_2
.= ((((ff . I) `1) ^2) / (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) + ((((ff . I) `2) ^2) / (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) by A150, SQUARE_1:def_2
.= ((((ff . I) `1) ^2) + (((ff . I) `2) ^2)) / (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)) by XCMPLX_1:62 ;
then  (((((ff . I) `1) ^2) + (((ff . I) `2) ^2)) / (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2))) * (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)) = 1 * (1 + ((((ff . I) `2) / ((ff . I) `1)) ^2)) by A134;
then  (((ff . I) `1) ^2) + (((ff . I) `2) ^2) = 1 + ((((ff . I) `2) / ((ff . I) `1)) ^2) by A150, XCMPLX_1:87;
then A151: ((((ff . I) `1) ^2) + (((ff . I) `2) ^2)) - 1 = (((ff . I) `2) ^2) / (((ff . I) `1) ^2) by XCMPLX_1:76;
 (ff . I) `1 <> 0 by A141, A143, A142, A144, A145, XREAL_1:64;
then  (((((ff . I) `1) ^2) + (((ff . I) `2) ^2)) - 1) * (((ff . I) `1) ^2) = ((ff . I) `2) ^2 by A151, XCMPLX_1:6, XCMPLX_1:87;
then A152: ((((ff . I) `1) ^2) - 1) * ((((ff . I) `1) ^2) + (((ff . I) `2) ^2)) = 0 ;
 (((ff . I) `1) ^2) + (((ff . I) `2) ^2) <> 0 by A144, COMPLEX1:1;
then A153: (((ff . I) `1) ^2) - 1 = 0 by A152, XCMPLX_1:6;
A154: |[(((gg . O) `1) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)))),(((gg . O) `2) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))))]| `2 = ((gg . O) `2) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) by EUCLID:52;
A155: 1 + ((((ff . O) `2) / ((ff . O) `1)) ^2) > 0 by Lm1;
 |[(((ff . O) `1) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)))),(((ff . O) `2) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))))]| `2 = ((ff . O) `2) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) by EUCLID:52;
then |.p1.| ^2 = ((((ff . O) `1) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)))) ^2) + ((((ff . O) `2) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)))) ^2) by A102, A129, A98, JGRAPH_1:29
.= ((((ff . O) `1) ^2) / ((sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) ^2)) + ((((ff . O) `2) / (sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)))) ^2) by XCMPLX_1:76
.= ((((ff . O) `1) ^2) / ((sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) ^2)) + ((((ff . O) `2) ^2) / ((sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) ^2)) by XCMPLX_1:76
.= ((((ff . O) `1) ^2) / (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) + ((((ff . O) `2) ^2) / ((sqrt (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) ^2)) by A155, SQUARE_1:def_2
.= ((((ff . O) `1) ^2) / (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) + ((((ff . O) `2) ^2) / (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) by A155, SQUARE_1:def_2
.= ((((ff . O) `1) ^2) + (((ff . O) `2) ^2)) / (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)) by XCMPLX_1:62 ;
then  (((((ff . O) `1) ^2) + (((ff . O) `2) ^2)) / (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2))) * (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)) = 1 * (1 + ((((ff . O) `2) / ((ff . O) `1)) ^2)) by A4;
then  (((ff . O) `1) ^2) + (((ff . O) `2) ^2) = 1 + ((((ff . O) `2) / ((ff . O) `1)) ^2) by A155, XCMPLX_1:87;
then A156: ((((ff . O) `1) ^2) + (((ff . O) `2) ^2)) - 1 = (((ff . O) `2) ^2) / (((ff . O) `1) ^2) by XCMPLX_1:76;
 (ff . O) `1 <> 0 by A126, A121, A127, A128, XREAL_1:64;
then  (((((ff . O) `1) ^2) + (((ff . O) `2) ^2)) - 1) * (((ff . O) `1) ^2) = ((ff . O) `2) ^2 by A156, XCMPLX_1:6, XCMPLX_1:87;
then A157: ((((ff . O) `1) ^2) - 1) * ((((ff . O) `1) ^2) + (((ff . O) `2) ^2)) = 0 ;
consider p3 being Point of (TOP-REAL 2) such that
A158: g . O = p3 and
A159: |.p3.| = 1 and
A160: p3 `2 <= p3 `1 and
A161: p3 `2 <= - (p3 `1) by A2;
A162: p3 <> 0. (TOP-REAL 2) by A159, TOPRNS_1:23;
A163: gg . O = (Sq_Circ ") . (g . O) by A8, FUNCT_1:12;
then A164: p3 = Sq_Circ . (gg . O) by A158, Th43, FUNCT_1:32;
A165: - (p3 `2) >= - (- (p3 `1)) by A161, XREAL_1:24;
then A166: (Sq_Circ ") . p3 = |[((p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2)))),((p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))))]| by A160, A162, Th30;
then A167: (gg . O) `2 = (p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) by A163, A158, EUCLID:52;
A168: sqrt (1 + (((p3 `1) / (p3 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A169: (gg . O) `1 = (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) by A163, A158, A166, EUCLID:52;
A170: now__::_thesis:_(_(gg_._O)_`2_=_0_implies_not_(gg_._O)_`1_=_0_)
assume  ( (gg . O) `2 = 0 & (gg . O) `1 = 0 ) ; ::_thesis: contradiction
then  ( p3 `2 = 0 & p3 `1 = 0 ) by A167, A169, A168, XCMPLX_1:6;
hence  contradiction by A162, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
 ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (- (p3 `2)) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) ) ) by A160, A165, A168, XREAL_1:64;
then A171: ( ( p3 `1 <= p3 `2 & (- (p3 `2)) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) ) or ( (gg . O) `1 >= (gg . O) `2 & (gg . O) `1 <= - ((gg . O) `2) ) ) by A167, A169, A168, XREAL_1:64;
then  ( ( (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) & - ((gg . O) `2) <= (gg . O) `1 ) or ( (gg . O) `1 >= (gg . O) `2 & (gg . O) `1 <= - ((gg . O) `2) ) ) by A163, A158, A166, A167, A168, EUCLID:52, XREAL_1:64;
then A172: Sq_Circ . (gg . O) = |[(((gg . O) `1) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)))),(((gg . O) `2) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))))]| by A167, A169, A170, Th4, JGRAPH_2:3;
A173: sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)) > 0 by Lm1, SQUARE_1:25;
A174: now__::_thesis:_not_(gg_._O)_`2_=_1
assume A175: (gg . O) `2 = 1 ; ::_thesis: contradiction
then  - (p3 `1) > 0 by A161, A164, A172, A154, A173, XREAL_1:139;
then  - (- (p3 `1)) < - 0 ;
hence  contradiction by A160, A164, A172, A173, A175, EUCLID:52; ::_thesis: verum
end;
 |[(((gg . O) `1) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)))),(((gg . O) `2) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))))]| `1 = ((gg . O) `1) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) by EUCLID:52;
then |.p3.| ^2 = ((((gg . O) `2) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)))) ^2) + ((((gg . O) `1) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)))) ^2) by A164, A172, A154, JGRAPH_1:29
.= ((((gg . O) `2) ^2) / ((sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) ^2)) + ((((gg . O) `1) / (sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)))) ^2) by XCMPLX_1:76
.= ((((gg . O) `2) ^2) / ((sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) ^2)) + ((((gg . O) `1) ^2) / ((sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) ^2)) by XCMPLX_1:76
.= ((((gg . O) `2) ^2) / (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) + ((((gg . O) `1) ^2) / ((sqrt (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) ^2)) by A125, SQUARE_1:def_2
.= ((((gg . O) `2) ^2) / (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) + ((((gg . O) `1) ^2) / (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) by A125, SQUARE_1:def_2
.= ((((gg . O) `2) ^2) + (((gg . O) `1) ^2)) / (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)) by XCMPLX_1:62 ;
then  (((((gg . O) `2) ^2) + (((gg . O) `1) ^2)) / (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2))) * (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)) = 1 * (1 + ((((gg . O) `1) / ((gg . O) `2)) ^2)) by A159;
then  (((gg . O) `2) ^2) + (((gg . O) `1) ^2) = 1 + ((((gg . O) `1) / ((gg . O) `2)) ^2) by A125, XCMPLX_1:87;
then A176: ((((gg . O) `2) ^2) + (((gg . O) `1) ^2)) - 1 = (((gg . O) `1) ^2) / (((gg . O) `2) ^2) by XCMPLX_1:76;
 (gg . O) `2 <> 0 by A167, A169, A168, A170, A171, XREAL_1:64;
then  (((((gg . O) `2) ^2) + (((gg . O) `1) ^2)) - 1) * (((gg . O) `2) ^2) = ((gg . O) `1) ^2 by A176, XCMPLX_1:6, XCMPLX_1:87;
then A177: ((((gg . O) `2) ^2) - 1) * ((((gg . O) `2) ^2) + (((gg . O) `1) ^2)) = 0 ;
 (((gg . O) `2) ^2) + (((gg . O) `1) ^2) <> 0 by A170, COMPLEX1:1;
then A178: (((gg . O) `2) ^2) - 1 = 0 by A177, XCMPLX_1:6;
 (((ff . O) `1) ^2) + (((ff . O) `2) ^2) <> 0 by A127, COMPLEX1:1;
then  (((ff . O) `1) ^2) - 1 = 0 by A157, XCMPLX_1:6;
hence  ( (ff . O) `1 = - 1 & (ff . I) `1 = 1 & (gg . O) `2 = - 1 & (gg . I) `2 = 1 ) by A131, A153, A148, A178, A174, A120, A123, Lm20; ::_thesis: verum
end;
then  rng ff meets rng gg by A2, A12, Th42, JGRAPH_1:47;
then A179: (rng ff) /\ (rng gg) <> {} by XBOOLE_0:def_7;
then  the Element of (rng ff) /\ (rng gg) in rng ff by XBOOLE_0:def_4;
then consider x1 being set  such that
A180: x1 in dom ff and
A181: the Element of (rng ff) /\ (rng gg) = ff . x1 by FUNCT_1:def_3;
 x1 in dom f by A180, FUNCT_1:11;
then A182: f . x1 in rng f by FUNCT_1:def_3;
 the Element of (rng ff) /\ (rng gg) in rng gg by A179, XBOOLE_0:def_4;
then consider x2 being set  such that
A183: x2 in dom gg and
A184: the Element of (rng ff) /\ (rng gg) = gg . x2 by FUNCT_1:def_3;
A185: gg . x2 = (Sq_Circ ") . (g . x2) by A183, FUNCT_1:12;
 x2 in dom g by A183, FUNCT_1:11;
then A186: g . x2 in rng g by FUNCT_1:def_3;
 ff . x1 = (Sq_Circ ") . (f . x1) by A180, FUNCT_1:12;
then  f . x1 = g . x2 by A181, A184, A1, A182, A186, A185, FUNCT_1:def_4;
then  (rng f) /\ (rng g) <> {} by A182, A186, XBOOLE_0:def_4;
hence  rng f meets rng g by XBOOLE_0:def_7; ::_thesis: verum
end;